Simulation method for estimating performance of product made of viscoelastic material

ABSTRACT

A simulation method includes the step of momently measuring a value of each of a strain, a strain speed, and a stress generated in the viscoelastic material, deriving time history data of a viscous drag, the strain speed and the stress, thereby deriving a relationship among the strain, the strain speed, and the viscous drag and setting the product as a product model whose performance is analyzed; inputting the relationship to the product model; and computing a stress and strain of a deviation component by using a deviation main strain and a deviation main strain speed converted from an entire coordinate system into a main strain coordinate system and a main strain speed coordinate system respectively to thereby conduct a simulation in consideration of a change of the viscous drag which occurs in dependence of a variation of the strain and the strain speed.

TECHNICAL FIELD

The present invention relates to a simulation method for estimating theperformance of a product composed of a viscoelastic material. Moreparticularly, the present invention relates to a simulation method foraccurately estimating the performance of the product composed of theviscoelastic material by means of a simulation.

BACKGROUND ART

A viscoelastic material represented by a macromolecular material such asrubber or elastomer is widely applied to various products such as tires,balls to be used in sports, rolls for printing machines.

In various products composed of the viscoelastic material or a metalmaterial, to save cost and time, development of products by using asimulation are made in various industrial fields. For example, toestimate the restitution performance of a golf ball, simulation methodsof actual ball-hitting tests are proposed.

In conducting the simulation, property values of a composing material ofa ball measured by a viscoelastic spectrum meter for measuring therigidity, the viscosity of the material and a tension testing machinefor measuring the modulus of longitudinal elasticity (Young's modulus)thereof are used as input data in the simulation. In particular, becausethe viscoelastic spectrum meter measures the property values of adynamic strain-applied specimen, the viscoelastic spectrum meter isuseful for simulating products composed of the viscoelastic material.

However in measurement conducted by using the viscoelastic spectrummeter and the tension testing machine for measuring the modulus oflongitudinal elasticity, a large deformation amount cannot be impartedto the specimen. Thus a maximum strain speed applied to the specimencomposed of the viscoelastic material at a measuring time is as low as0.001/s to 1.0/s and a maximum compression strain is also as low as0.0001 to 0.02.

A product composed of the viscoelastic material may deform at a highspeed and greatly owing to the influence of an external force appliedthereto when it is actually used. For example, when the golf ball ishit, a maximum strain speed of a material for the golf ball is as highas 500/s to 5000/s, and a maximum compression strain thereof is as largeas 0.05-0.50.

As described above, the viscoelastic spectrum meter and the tensiontesting machine for measuring the modulus of longitudinal elasticity areincapable of measuring the property values of the viscoelastic materialin a condition equivalent to a condition where the product composed ofthe viscoelastic material deforms quickly and greatly when it isactually used. Thus the maximum strain speed of the viscoelasticmaterial and its maximum compression strain measured at a simulationtime are much different from those measured at the time when the productcomposed of it is actually used. Therefore the conventional simulationmethod of inputting the property value obtained by using theviscoelastic spectrum meter and the tension testing machine is incapableof accomplishing an accurate simulation by taking the property of theviscoelastic material into consideration.

That is, it is known that the deformation behavior of the viscoelasticmaterial when an impact load is applied thereto is different from thatof the viscoelastic material when a static load is applied thereto. Thatis, the deformation behavior of the viscoelastic material is greatlyinfluenced by a deformation amount or a deformation speed. Inparticular, when a macromolecular material such as rubber and elastomeris subjected to the impact load, it deforms at a speed as high asseveral seconds by 10000 or several seconds by 1000 and as greatly as byseveral tens of percentages in a quantitative respect. There are manyviscoelastic materials that deform at such a high speed and in such alarge amount. To develop products efficiently, there is a demand fordevelopment of a simulation method capable of conducting an accuratesimulation. More specifically, the performance of a product such as thegolf ball to which an impact is applied when it is used depends on adynamic behavior in a condition where it deforms at a high speed andgreatly. The performance of the product also depends on thecharacteristic of the material thereof. Therefore to develop a product,it is indispensable to conduct an accurate simulation in a conditionequivalent to a condition in which the product composed of the materialis actually used.

Some viscoelastic materials change in its property value such as itsrigidity (modulus of longitudinal elasticity and modulus of transverseelasticity) and loss coefficient in dependence on the magnitude of astrain and a strain speed when an external force such as an impact loadis applied thereto. That is, the viscoelastic material is diverse in itsdeformation speed and deformation magnitude. Thus depending on thedeformation speed and the deformation magnitude, the viscoelasticmaterial has a property that it changes not linearly but highlynonlinearly. More specifically, as the viscoelastic material is deformedby an external force applied thereto and strained increasingly, the looparea of an S-S (strain-stress) curve increases, and the property such asthe loss coefficient thereof changes in dependence on a deformationstate (speed and magnitude of deformation) thereof, thus showingnonlinearity in its property. Many viscoelastic materials have a highnonlinearity in their properties. Thus there is a demand for developmentof a simulation method capable of simulating a product composed of sucha viscoelastic material.

However there are no methods capable of accurately expressing aphenomenon that the property of the viscoelastic material, for example,its rigidity (modulus of longitudinal elasticity and modulus oftransverse elasticity) and loss coefficient changes nonlinearly in ahigh extent in dependence on the deformation speed and deformationmagnitude thereof. Simulations have been hitherto conducted on theassumption that the property value of the viscoelastic materialcomposing the golf ball or the like hardly changes. Consequently theconventional simulation method has a disadvantage that it is incapableof correctly estimating the performance of the product composed of theviscoelastic material in an actual use. Thus to estimate the performanceof the product, trial manufacture cannot but be made.

The present invention has been made in view of the above-describedsituation. Thus, it is an object of the present invention to accuratelyestimate the performance of a product composed of a viscoelasticmaterial showing nonlinearity in its property, for example, a productcomposed of a viscoelastic material whose rigidity such as the modulusof longitudinal elasticity changes in dependence on a magnitude of astrain and that of a strain speed, by conducting a simulation in acondition in which the product is actually used.

DISCLOSURE OF THE INVENTION

To solve the above-described problem, according to the presentinvention, firstly, there is provided a simulation method of estimatingperformance of a product made of a viscoelastic material, comprising thesteps of momently measuring a value of each of a strain, a strain speed,and a stress generated in the viscoelastic material in a measuringcondition equivalent to a condition in which the product composed of theviscoelastic material is actually used; deriving time history data of aviscous drag of the viscoelastic material from time history data of eachof the strain, the strain speed, and the stress and a viscoelastic modelin which a viscosity of the viscoelastic material is considered, therebyderiving a relationship among the strain, the strain speed, and theviscous drag;

-   -   in estimating the performance of the product model whose        performance is analyzed and made of the viscoelastic material,        the product model is divided into a large number of elements,        said relationship is inputted to the product model, and an        analysis is executed by a finite element method in consideration        of a change of the viscous drag in dependence on a variation of        the strain and the strain speed; and resolving the strain and        the strain speed of an entire coordinate system generated in        each element into deviation components and volume components,        and converting the strain and the strain speed of the deviation        components from the entire coordinate system into a main strain        coordinate system and a main strain speed coordinate system        respectively; and determining a viscous drag for a coordinate        axis of each of the main strain coordinate system and the main        strain speed coordinate system by using a converted deviation        main strain and a converted deviation main strain speed.

As described above, in the first invention, the strain and the strainspeed measured in the condition in which the product is actually usedare resolved into the deviation components and the volume components.Then by using the main strain and the main strain speed of the deviationcomponents converted into the main strain coordinate system and the mainstrain speed coordinate system respectively, the viscous drag isdetermined for the coordinate axis of each of the main strain coordinatesystem and the main strain speed coordinate system in such a way thatfor each element, the viscous drag is variable for three components inthe direction of each axis. Therefore it is possible to accomplish acorrect analysis of a three-dimensional direction and for theviscoelastic material whose viscous drag changes in dependence on adeformation state (magnitude of strain and strain speed). Thus it ispossible to improve the estimation precision of the simulation.

In the first invention, it is possible to accomplish the simulation ofthe material whose loss coefficient shows a high nonlinearity with highprecision and estimate the performance of the product, assuming that theproduct is in an actual use state.

In the first invention, to consider the viscoelastic characteristic inthe deviation components of the strain and the strain speed generated ineach element in the entire coordinate system having components ε_(x),ε_(y), ε_(z), ε_(xy), ε_(yz), ε_(zx), the strain and the strain speedare resolved into the deviation components and the volume components.Then by using the main strain and the main strain speed of the deviationcomponents converted into the main strain coordinate system and the mainstrain speed coordinate system respectively, the viscous drag isdetermined for the coordinate axis of each of the main strain coordinatesystem and the main strain speed coordinate system in such a way thatfor each element, the viscous drag is variable for three components inthe direction of each axis. Therefore it is possible to make a correctanalysis of a three-dimensional direction in consideration of theviscoelasticity of the viscoelastic material and improve the simulationprecision.

To solve the above-described problem, in the invention, secondly, thereis provided a simulation method of estimating performance of a productmade of a viscoelastic material, comprising the steps of momentlymeasuring a value of each of a strain, a strain speed, and a stressgenerated in the product made of a viscoelastic material in a measuringcondition equivalent to a condition in which the product composed of theviscoelastic material is actually used; computing a plurality ofdifferent rigidities from time history data of each of the strain andthe stress and deriving a correspondence relationship among the strain,the strain speed, and the rigidities; in estimating the performance ofthe product model made of a viscoelastic material by setting the productmade of the viscoelastic material as a product model whose performanceis analyzed, the product model is divided into a large number ofelements, the relationship among the strain, the strain speed, and therigidities is inputted to the product model, and an analysis is executedby a finite element method in consideration of a change of therigidities in dependence on a variation of the strain and the strainspeed; and resolving the strain and the strain speed of an entirecoordinate system generated in each element into deviation componentsand volume components, and converting the strain and the strain speed ofthe deviation components from the entire coordinate system into a mainstrain coordinate system and a main strain speed coordinate systemrespectively; and determining a rigidity for a coordinate axis of eachof the main strain coordinate system and the main strain speedcoordinate system by using a converted deviation main strain and aconverted deviation main strain speed.

As described above, in the second invention, the strain and the strainspeed measured in the condition in which the product is actually usedare resolved into the deviation components and the volume components.Then by using the main strain and the main strain speed of the deviationcomponents converted into the main strain coordinate system and the mainstrain speed coordinate system respectively, the rigidity (modulus oflongitudinal elasticity or/and modulus of transverse elasticity) isdetermined for the coordinate axis of each of the main strain coordinatesystem and the main strain speed coordinate system in such a way thatfor each element, the rigidity is variable for three components in thedirection of each axis. Therefore it is possible to accomplish a correctanalysis of a three-dimensional direction and for the viscoelasticmaterial whose rigidity changes in dependence on a deformation state(magnitude of strain and strain speed). Thus it is possible to improvethe estimation precision of the simulation.

In the second invention, it is possible to accomplish the simulation ofthe material whose modulus of longitudinal elasticity shows a highnonlinearity with high precision and estimate the performance of theproduct, assuming that the product is in an actual use state.

It is preferable to derive the time history data of the viscous drag ofthe viscoelastic material from the time history data of each of thestrain, the strain speed, and the stress and from the viscoelastic modelin which the viscosity of the viscoelastic material is considered andconsider the change of the viscous drag as well as the rigidity such asthe modulus of longitudinal elasticity and the modulus of transverseelasticity. By considering the change of the viscous drag as well as therigidity such as the modulus of longitudinal elasticity and the modulusof transverse elasticity in dependence on a deformation state of thematerial and the change of the viscous drag, it is possible toaccomplish a correct analysis of the material showing the nonlinearity.

In the second invention, to consider the viscoelastic characteristic inthe deviation components of the strain and the strain speed generated ineach element in the entire coordinate system having components ε_(x),ε_(y), ε_(z), ε_(xy), ε_(yz), ε_(zx), the strain and the strain speedare resolved into the deviation components and the volume components.Then by using the main strain and the main strain speed of the deviationcomponents converted into the main strain coordinate system and the mainstrain speed coordinate system respectively, the viscous drag and therigidity (modulus of longitudinal elasticity or/and modulus oftransverse elasticity) are determined for the coordinate axis of each ofthe main strain coordinate system and the main strain speed coordinatesystem in such a way that for each element, the viscous drag and therigidity are variable for three components in the direction of eachaxis. Therefore it is possible to make a correct analysis of athree-dimensional direction in consideration of the viscoelasticity ofthe viscoelastic material and improve the simulation precision.

To solve the above-described problem, in the invention, thirdly, thereis provided a simulation method of estimating performance of a productmade of a viscoelastic material, comprising the steps of momentlymeasuring a value of each of a strain, a strain speed, and a stressgenerated in the product made of a viscoelastic material in a measuringcondition equivalent to a condition in which the product composed of theviscoelastic material is actually used; deriving time history data of aviscous drag of the viscoelastic material separately in a strainincrease state and a strain decrease state or a strain restoration statefrom time history data of each of the strain, the strain speed, and thestress and a viscoelastic model in which a viscosity of the viscoelasticmaterial is considered, thereby deriving a relationship among thestrain, said strain speed, and said viscous drag;

-   -   in estimating the performance of the product model whose        performance is analyzed and made of the viscoelastic material,        the product model is divided into a large number of elements,        said relationship is inputted to the product model, and an        analysis is executed by a finite element method in consideration        of a difference in the viscous drag between the strain increase        state and the strain decrease state or the strain restoration        state; and resolving the strain and the strain speed of an        entire coordinate system generated in each element into        deviation components and volume components, and converting the        strain and the strain speed of the deviation components from the        entire coordinate system into a main strain coordinate system        and a main strain speed coordinate system respectively; and        determining a viscous drag different between the strain increase        state and the strain decrease or the strain restoration state        for a coordinate axis of each of the main strain coordinate        system and the main strain speed coordinate system when strains        having an equal value are generated in the viscoelastic        material, by using a converted deviation main strain and a        converted deviation main strain speed.

As described above, in the third invention, the strain and the strainspeed measured in the condition in which the product is actually usedare resolved into the deviation components and the volume components.Then by using the main strain and the main strain speed of the deviationcomponents converted into the main strain coordinate system and the mainstrain speed coordinate system respectively, the viscous drag differentbetween the strain increase state and the strain decrease or the strainrestoration state is determined for the coordinate axis of each of themain strain coordinate system and the main strain speed coordinatesystem in such a way that for each element, the viscous drag is variablefor three components in the direction of each axis, when strains havingan equal value are generated. Therefore it is possible to accomplish aprecise analysis in correspondence to an actual situation, namely, independence on the strain increase state and the strain decrease or thestrain restoration state and accomplish a precise correct analysis of athree-dimensional direction. Thus it is possible to improve theestimation precision of the simulation.

In the third invention, it is possible to accomplish the simulation ofthe material whose loss coefficient shows a high nonlinearity with highprecision and estimate the performance of the product in an assumptionof the condition in which the product is actually used.

That is, the deformation state generated in the viscoelastic materialcan be divided into the strain increase state in which the strainincreases in a compression direction and the strain restoration state inwhich a compression amount decreases gradually. In dependence on thestrain increase state and the strain decrease state, the relationshipbetween the strain and the strain speed varies. Therefore it is possibleto accurately grasp the deformation state of an actual material byconsidering the two states of the strain generated in the viscoelasticmaterial, namely, the strain increase state and the strain decreasestate (or strain restoration state), determining the viscous drag byusing the relationship, between the strain and the strain speed,different in the strain increase state and the strain decrease state,and deriving a viscous drag value separately in the strain increasestate and the strain decrease state when strains having an equal valueare generated. Therefore it is possible to express a phenomenon in whichthe material changes according to a deformation speed thereof and a sizethereof and improve the simulation precision.

It is preferable to compute a norm which is the magnitude of a mainstrain for each of the elements, and compare the computed norm with anorm of a previous step in a simulation to determine that the mainstrain is in the increase state when the norm has increased, and todetermine that the main strain is in the decrease state when the normhas decreased.

For example, a slight change in the main strain in only one directioncauses the strain increase state and the strain decrease state to havelarge variations. However, when the main strain has magnitudes in amulti-direction, the strain increase state and the strain decrease stateare not changed unless there are comparatively large variations in themain strain. Thus it is possible to enhance stability in computations.

In the third invention, to consider the viscoelastic characteristic inthe deviation components of the strain and the strain speed generated ineach element in the entire coordinate system having components ε_(x),ε_(y), ε_(z), ε_(xy), ε_(yz), ε_(zx), the strain and the strain areresolved into the deviation components and the volume components. Thenby using the main strain and the main strain speed of the deviationcomponents converted into the main strain coordinate system and the mainstrain speed coordinate system respectively, the viscous drag differentbetween the strain increase state and the strain decrease state or thestrain restoration state is determined for the coordinate axis of eachof the main strain coordinate system and the main strain speedcoordinate system in such a way that for each element, the viscous dragis variable for three components in the direction of each axis.Therefore it is possible to make a correct analysis of athree-dimensional direction in consideration of the viscoelasticity ofthe viscoelastic material and improve the simulation precision.

In dependence on a case, the viscous drag is determined separately in astrain increase state of the stress and a strain decrease state thereof.

In the simulation method of the first, second, and third inventions ofthe present invention, the volume components mean a component expressedby ε_(v)=(ε_(x)+ε_(y)+ε_(z)), and the deviation components meancomponents expressed by ε_(xy)′=ε_(xy)/2, ε_(yz)′=ε_(yz)/2,ε_(zx)′=ε_(zx)/2, ε_(x)′=ε_(x)−ε_(v)/3, ε_(y)′=ε_(y)−ε_(v)/3, andε_(z)′=ε_(z)−ε_(v)/3. The reason the shear component is multiplied by ½is because a physical strain is converted.

The reason only the deviation component is converted is because theviscoelastic component of the volume component is very small and cannotbe found by measurement in a current situation. That is, since thevolume component has little viscoelastic component, determination ismade by only the deviation component. Since from the result of themeasurement, the conversion has a result in the direction of only oneaxis, a conversion of ε_(xy), ε_(yz), ε_(zx) components=0 is made. Theresult of measurement is used for only the three components ε₁, ε₂,ε₃(three components of main strain).

The entire coordinate system means a coordinate system in which aninitial configuration, namely, the configuration of a model isdetermined. The main strain coordinate system and the main strain speedcoordinate system mean a coordinate system in which when the strain andthe strain speed are converted in such a way that a shear component iszero, the entire coordinate system is converted at the same angle.

The strain (strain speed) generated in each element consists of sixcomponents. Because the property of the material measured in thecondition of a high speed and a large deformation shows the propertythereof in the direction of one axis, consideration for the propertythereof in a shear direction is insufficient. Therefore the sixcomponents of the strain generated in each element are converted intothe main strain coordinate system to determine the three components ofthe strain in the direction of a main axis.

That is, by making the strain in the shear direction zero, the sixcomponents (ε_(x), ε_(y), ε_(z), ε_(xy), ε_(yz), ε_(zx)) of the entirecoordinate system is reduced to three components of the strain (mainstrain coordinate system, ε₁, ε₂, ε₃) in the direction of the main axis.Thereby without considering the shear direction, results of measurementallows evaluation of the characteristic of the viscoelastic material.

More specifically, the strain and the strain speed of the entirecoordinate system generated in each element are resolved into the volumecomponents and the deviation components in which the viscoelasticity istaken into consideration. Then the six deviation components of theresolved strain and those of the strain speed are converted from theentire coordinate system into the main strain coordinate system and themain strain speed coordinate system respectively. Thereafter threecomponents of the deviation main strain in the main strain coordinatesystem and three components of the deviation main strain speed in themain strain speed coordinate system are used. Thereby the viscous dragand the rigidity (modulus of longitudinal elasticity and modulus oftransverse elasticity) can be determined for each coordinate axis ofeach of the main strain coordinate system and the main strain speedcoordinate system.

The stress of the deviation component and the strain thereof arecomputed by considering the viscoelasticity from the viscous drag.Thereafter the stress of the deviation component and the strain thereofare re-converted into those of the entire coordinate system. Based onthe stress of the volume component and the strain thereof and the stressof the deviation component and the strain thereof, the stress and strainof each element are computed. By computing the stress and strain of eachelement as described above, the stress and the strain can be computedmomently.

More specifically, from equations 1 and 2 shown below and by using thedeviation main strain and the deviation main strain speed, the viscousdrag, the rigidity (modulus of longitudinal elasticity and modulus oftransverse elasticity), and the viscous drag different between thestrain increase state and the strain decrease state or the strainrestoration state are determined for each coordinate axis as necessary,and the stress and strain of the deviation component are computed byconsidering the viscoelasticity from the viscous drag.(Equation 1) $\begin{matrix}{E_{l}^{n + 1} = {{E_{i}^{n}\quad{\exp\left( {{- \beta_{i}}{dt}} \right)}} - {\frac{1 - {\exp\left( {{- \beta_{i}}{dt}} \right)}}{\beta_{i}}{\overset{.}{\gamma}}_{i}}}} & (1)\end{matrix}$Where E_(i): deviation main strain of an i-th element, n: step ofanalysis time,β_(i): β(β=E_(t)/η, E_(t)=E_(short)−E_(long), η: viscous drag) of thei-th element, andγ_(i): deviation main strain speed of the i-th element(Equation 2)Δσ_(i) =F _(i) G _(short){dot over (γ)}_(i)+2F _(i)(G _(short) −G_(long))(E _(i) ^(n+1) −E _(i) ^(n))   (2)

-   -   where σ_(i): stress of the i-th element, F_(i): rigidity        coefficient, G_(short): short-term shear coefficient, and        G_(long): long-term shear coefficient

After β in the equations 1 and 2 and the rigidity coefficient F arecomputed from the experimental data, the six deviation components of thestrain are updated by using the equation 1. Then the stress is computedby using the equation 2. After the main strain coordinate system and themain strain speed coordinate system are converted into the entirecoordinate system, the stress is updated by using the following equation3:

(Equation 3)σ_(ij)=Δσ_(ij) +KΔε _(v)   (3)

-   -   where K: modulus of elasticity of volume, ε_(v): volume strain,        σ_(ij): stress of entire coordinate system

Further, by using a viscoelastic model in which the viscosity of theviscoelastic material is considered, the viscous drag, the rigidity suchas the modulus of longitudinal elasticity, and the viscous dragdifferent between the strain increase state and the strain decreasestate or the strain restoration state are derived as necessary. Then therelationship among the strain, the strain speed, the viscous drag; therelationship among the strain, the strain speed, the viscous drag, andthe rigidity such as the modulus of longitudinal elasticity; and therelationship among the strain, the strain speed, and the viscous dragdifferent between the strain increase state and the strain decreasestate or the strain restoration state are inputted to the product modelto execute the simulation.

Therefore it is possible to accurately express the phenomenon that thevalue of the property of the viscoelastic material showing nonlinearlychanges in dependence on its deformation speed and magnitude. Furtherbecause the value of each of the strain, the strain speed, and thestress is measured in the condition equivalent to the condition in whichthe product composed of the viscoelastic material is actually used, itis possible to conduct the simulation in correspondence to variousdeformation states of the viscoelastic material. Accordingly by thesimulation, it is possible to accurately estimate the performance of theproduct composed of the viscoelastic material in which the relationshipbetween the strain and the strain speed changes in dependence on adeformation state and whose property such as the loss coefficient andthe modulus of longitudinal elasticity shows nonlinearity.

In the simulation method of the present invention, the value of each ofthe strain, the strain speed, and the stress generated in the productcomposed of the viscoelastic material is measured momently in ameasuring condition equivalent to the condition in which the productcomposed of the viscoelastic material is actually used. Morespecifically, the measuring condition is set on the assumption of thestate in which an external force is applied to the product in an actualuse state and the viscoelastic material deforms. The value of each ofthe strain, the strain speed, and the stress generated in theviscoelastic material is measured momently in the above-describedmeasuring condition to obtain the time history data of each of thestrain, the strain speed, and the stress. Thus it is possible to obtainthe information of a deformation state of the viscoelastic material,when an external force is applied to the product supposed to be in anactual use state. Thereby it is possible to correctly estimate theproperty of the viscoelastic material which deforms greatly and quicklyby an impact load.

It is preferable to measure the value of each of the strain, the strainspeed, and the stress momently in a plurality of measuring conditions.By altering the magnitude of the external force applied to the productand setting a plurality of measuring conditions, it is possible toobtain data of various patterns about the strain, the strain speed, andthe stress and improve the accuracy of input values in the simulation.To obtain data of patterns as much as possible, it is preferable tomeasure the value of each the strain, the strain speed, and the stressfrom a time when the strain is generated in the viscoelastic materialupon application of an external force thereto until the strain becomesapproximately zero. It is also preferable to perform measurement atshort intervals to conduct the simulation with high precision.

It is preferable to compose the viscoelastic model of a spring and adashpot in view of the viscosity of the viscoelastic material. Becausesuch a viscoelastic model simplifies the viscosity of the viscoelasticmaterial, it is easy to consider the influence of the viscosity on adeformation state of the viscoelastic material. More specifically, amaxwell model, Voight model, and a combination of a plurality of springsand dashpots are favorable. To simplify the construction of theviscoelastic model, a two-component model is favorable. Theseviscoelastic material models are used in such a way that the viscousdrag of the dashpot and the rigidity of the spring (modulus oflongitudinal elasticity E or shear coefficient (modulus of transverseelasticity)) are variable. The shear coefficient is a property valuedetermined by the modulus of longitudinal elasticity (Young's modulus) Eand Poisson's ratio.

The time history data of the viscous drag of the viscoelastic materialand the time history data of the viscous drag different between thestrain increase state and the strain decrease state or the strainrestoration state are derived from the time history data of each of thestrain, the strain speed, and the stress and from the viscoelastic modelin which the viscosity of the viscoelastic material is considered.

More specifically, the relationship among the strain, the strain speed,and the viscous drag; the relationship among the strain, the strainspeed, the viscous drag, and the rigidity such as the modulus oflongitudinal elasticity; the relationship among the strain, the strainspeed, and the viscous drag different between the strain increase stateand the strain decrease state or the strain restoration state areestablished as equations from the viscoelastic model. In this manner,the viscous drag is expressed as the function of the strain and thestrain speed. The modulus of longitudinal elasticity of the viscoelasticmaterial is derived in dependence on the strain and the strain speed,generated therein, obtained by the measurement. The value of the viscousdrag different between the strain increase state and the strain decreasestate or the strain restoration state is derived by substituting thederived modulus of longitudinal elasticity, the strain, and the strainspeed into the function.

The rigidity such as the modulus of longitudinal elasticity isdetermined in dependence on the strain and the strain speed. The viscousdrag is expressed as the relation among the strain, the strain speed,and the rigidity such as the modulus of longitudinal elasticity. Thevalue of the viscous drag can be also derived in consideration of achange of the rigidity such as the modulus of longitudinal elasticity bysubstituting the strain, the strain speed, and the rigidity such as themodulus of longitudinal elasticity into the function.

Since the time history data of each of the strain, the strain speed, andthe stress is obtained by the measurement, the time history data of theviscous drag corresponding to the strain and the strain speed can beobtained. As described above, the viscous drag is determined independence on the value of the strain and the strain speed and thuschanges in correspondence to a variation of the strain and the strainspeed which is made with the elapse of time.

The relationship among the strain, the strain speed, and the viscousdrag; the relationship among the strain, the strain speed, the viscousdrag, and the rigidity such as the modulus of longitudinal elasticity;the relationship among the strain, the strain speed, and the viscousdrag different between the strain increase state and the strain decreasestate or the strain restoration state are inputted to computation inputdata (or input data) including information of the product, composed ofthe viscoelastic material, set as a product model whose performance isanalyzed. The computation input data also includes speeds andrestriction conditions. Thereby a simulation is conducted inconsideration of a change of the viscous drag and a change of therigidity such as the modulus of longitudinal elasticity in dependence ona variation of the strain and the strain speed.

The relationship among the strain, the strain speed, and the viscousdrag different between the strain increase state and the strain decreasestate or the strain restoration state is inputted to the product modelin computations for the simulation. More specifically, two-dimensionaldata of the relationship between the strain and the strain speed and therelationship between the strain and the viscous drag are inputted to theproduct model to perform computations. It is possible to generatethree-dimensional data of the relationship among the strain, the strainspeed, and the viscous drag and input the data to the product model asthe function of the strain and the strain speed to perform computations.

As the two-dimensional data of the relationship between the strain andthe strain speed and the relationship between the strain and the viscousdrag, the strain, the strain speed, and the viscous drag correspondingto the strain as well as the strain speed and different between thestrain increase state and the strain decrease state or the strainrestoration state can be written as input data by using theabove-described relationships. By using the above-describedrelationships, the strain, the strain speed, the rigidity such as themodulus of longitudinal elasticity corresponding to the strain as wellas the strain speed, and the viscous drag corresponding to the strain aswell as the strain speed can be written as input data.

More specifically, the strain, the strain speed, and the like aremeasured in a plurality of measuring conditions. Regarding each ofpatterns having different measuring conditions, the correspondencerelationship between the strain and the strain speed is recorded fromtime series data of the strain and the strain speed. The value of theviscous drag and that of the rigidity corresponding to each curve arealso recorded. By properly adjusting the correspondence relationshipamong the strain, the strain speed, and the viscous drag, the viscousdrag at a given strain and a given strain speed is accurately derived toperform computations.

The more the number of measurements in different conditions of thestrain and strain speed is, the more accurately the property of theviscoelastic material can be materialized. Thus it is preferable tomeasure the strain, the strain speed, and the like in a plurality ofdifferent measuring conditions. However the more the number of data ofmeasurement is, the more it takes to perform computations for conductingthe simulation. In the case of a strain and a strain speed not the sameas the data of the strain and the strain speed measured under apredetermined measuring condition, it is preferable to compute theviscous drag and the rigidity by using an interpolation. As theinterpolation, a primary interpolation is performed by using a binaryvalue of the viscous drag or that of the modulus of longitudinalelasticity determined in dependence on a close strain and a close strainspeed or an interpolation which is performed by using each of valuesmeasured in all measuring conditions. By performing such aninterpolating operation, it is possible to compute the viscous drag andthe modulus of longitudinal elasticity in correspondence to variationsof the strain and the strain speed generated in the viscoelasticmaterial which is made according to measuring conditions.

The simulation method of the present invention allows a correctsimulation of the property and deformation behavior of the viscoelasticmaterial showing nonlinearity in its property in the conditionequivalent to the condition in which the product composed of theviscoelastic material is actually used. By using the viscoelastic modelwhose viscous drag and the modulus of longitudinal elasticity aredetermined by the value of the strain and the strain speed and computingthe viscous drag, it is possible to consider the nonlinearity of theviscoelastic material which is not linear in its property but deformsaccording to its deformation speed and deformation magnitude.

The simulation method of the present invention is carried out by afinite element method described below.

In conducting the simulation method by the finite element method, alarge number of nodal points and elements are set on the product model.That is, in estimating the property of the viscoelastic materialcomposing the product by simulating the product by the finite elementmethod, the rigidity (modulus of longitudinal elasticity or shearcoefficient) of the spring composing the viscoelastic model isdetermined, and the viscous drag showing viscosity is determined by thestrain and the strain speed generated in each element, as shown above.Thereby for each element, it is possible to indicate the property of theviscoelastic material in a proper condition of the strain and the strainspeed. Instead of inputting the modulus of longitudinal elasticity, ashear coefficient may be inputted in relation to the Poisson's ratio.Whether the modulus of longitudinal elasticity or the shear coefficientis used depends on the specification of the program of the finiteelement method. The simulation method of the present invention can bealso applied to the case where not Young's modulus but the shearcoefficient is determined in an impact compression test.

The strain, the strain speed, and the stress are measured by a splitHopkinson rod testing machine. The split Hopkinson rod testing machineis capable of giving a high-speed and large-deformation strain to aspecimen. That is, owing to the use of the split Hopkinson rod testingmachine, it is possible to obtain time series data of each of thestrain, the strain speed, and the stress of the viscoelastic material ina high-speed and a large-deformation condition in which the viscoelasticmaterial deforms as high as several seconds by 10000 or several secondsby 1000 and as great as by several tens of percentages in a quantitativerespect. Assuming that the measuring condition of the split Hopkinsonrod testing machine is equivalent to the condition of the strain and thestrain speed generated in the viscoelastic material when an impact loadis applied to a product, it is possible to obtain property of theviscoelastic material corresponding to various states such as the statein which the viscoelastic material deforms very quickly and greatly.Thus by using the property measured by the split Hopkinson rod testingmachine, the accuracy of the simulation can be improved.

The split Hopkinson rod testing machine is capable of measuring propertyof the material composing the specimen in various regions of the strainand the strain speed by only altering the collision speed of its hittingrod which applies an impact to the specimen. Therefore the splitHopkinson rod testing machine makes it possible to obtain the propertyof the material in various strains and strain speeds.

The split Hopkinson rod testing machine is originally used to evaluatean impact behavior of a metal material. In the present invention, thesplit Hopkinson rod testing machine is improved in its construction toevaluate the viscoelastic material having viscosity. The method ofmeasuring the property of a material by using the split Hopkinson rodtesting machine will be described later.

Needless to say, the property of the material may be measured by ameasuring method other than the method carried out by using the splitHopkinson rod, provided that the measuring method is capable of giving ahigh-speed and large-deformation strain to a specimen and that theproperty of the material of the specimen can be measured in a measuringcondition equivalent to a condition in which a product composed of thematerial is actually used.

When a strain, a strain speed, and a stress generated in theviscoelastic material are measured in a measuring condition equivalentto the condition in which the product composed of the viscoelasticmaterial is actually used, a maximum value of the strain generated inthe viscoelastic material is in the range of 0.05 to 0.50 and/or amaximum value of the strain speed is in the range of 500/s to 10000/sand favorably in the range of 500/s to 5000/s. The range of the maximumvalue of the strain and that of the strain speed are the condition ofthe strain and the strain speed generated when the viscoelastic materialdeforms at a high speed and greatly. Thus to estimate the performance ofthe product when it deforms at a high speed and greatly, it ispreferable to use time series data of each of the strain, the strainspeed, and the stress in this condition.

The viscoelastic material is used for a golf ball. The product model isa golf ball. The golf ball is a product composed of the viscoelasticmaterial and deforms at a high speed or in a large amount upon receiptof an external force such as an impact load, when the golf ball isactually used. The state of the golf ball deforming at a high speed andin a large amount affects the performance thereof greatly. Therefore theanalysis based on the simulation method of the present invention is veryuseful for estimating the performance of the golf ball. The simulationmethod is capable of estimating the performance of the golf ball withhigh accuracy without making a trial manufacture. Thus the simulationmethod allows efficient designing of the golf ball.

A phenomenon of a collision between the golf ball and a hitting objectassumed to be a golf club head is simulated to estimate a behavior ofthe golf ball at the time of the collision. The simulation method iscapable of estimating the property of the viscoelastic materialcomposing the golf ball in a condition of a strain, a strain speed, anda stress equivalent to those generated in the viscoelastic material ofthe golf ball when the golf ball is actually hit with the golf clubhead. Therefore simulation method is capable of estimating therestitution coefficient of the golf ball and the behavior thereof suchas a deformation state thereof when it is hit.

The simulation method of the present invention is applicable to aso-called one-piece golf ball composed of a crosslinked rubber layer, aso-called two-piece golf ball composed of a core made of a crosslinkedrubber layer and a cover covering the core, and a so-called multi-piecegolf ball composed of three or more layers. That is, the simulationmethod of the present invention is applicable to golf balls composed ofthe viscoelastic material and having any constructions.

The viscoelastic material includes materials having viscosity. Forexample, thermoplastic resin, thermosetting resin, elastomers, andrubber can be used as the viscoelastic material. It is possible to usethese viscoelastic materials singly or a mixture thereof. It is possibleto add additives such as a colorant, a deterioration prevention agent,and a crosslinking agent to these viscoelastic materials and the mixturethereof as necessary. The simulation method is applicable to allmaterials so long as the strain, the strain speed, and the stressthereof can be measured in a condition where a product made of them isactually used.

As the viscoelastic material, it is possible to use synthetic resin suchas ionomer which is used as a material for a golf ball, polybutadiene(butadiene rubber), natural rubber, polyisoprene, styrene-butadienecopolymer, ethylene-propylene-diene copolymer (EPDM), and urethanerubber.

As products composed of the viscoelastic material, in addition to thegolf ball, a rubber roller and a rubber belt for a printing apparatus, atire, and sports goods, for example, goods for tennis, golf, and thelike are known. It is necessary that the viscoelastic material is usedin at least one portion of a product. For example, the viscoelasticmaterial may be used in combination with other materials such as a metalmaterial to form a composite molded product. The simulation method iscapable of estimating the performance of a portion, of the product,constructed of the viscoelastic material. The simulation method ispreferably applicable to a product that deforms at a high speed and in alarge amount when it is subjected to an impact load. The simulationmethod is capable of estimating the performance of the product and itsdynamic behavior with high accuracy.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows time history data of a strain ε in a first mode of thepresent invention of the present invention.

FIG. 2 shows time history data of a strain speed ε′ in the first mode ofthe present invention.

FIG. 3 shows time history data of a stress σ in the first mode of thepresent invention.

FIG. 4 shows a strain-stress curve and a method of computing a modulusof longitudinal elasticity in the first mode of the present invention.

FIG. 5 shows a two-component Voight model used as a viscoelastic modelof the first mode of the present invention.

FIG. 6 shows time history data of a viscous drag in the first mode ofthe present invention.

FIG. 7 shows the relationship between the strain and the viscous drag inthe first mode of the present invention.

FIG. 8 shows the relationship between the strain and the strain speed inthe first mode of the present invention.

FIG. 9 shows a division situation of a golf ball model by means ofmeshes.

FIG. 10 shows a situation of a collision between a hollow rod model madeof aluminum and a golf ball model.

FIG. 10A shows a situation before the collision.

FIG. 10B shows a situation at the time of the collision.

FIG. 10C shows a situation after the collision.

FIG. 11 shows a method of computing a loss coefficient.

FIG. 12 shows time history data of a strain ε in a second mode of thepresent invention.

FIG. 13 shows time history data of a strain speed ε′ in the second modeof the present invention.

FIG. 14 shows time history data of a stress σ in the second mode of thepresent invention.

FIG. 15 shows a strain-stress curve and a method of computing a modulusof longitudinal elasticity in the second mode of the present invention.

FIG. 16 shows time history data of a viscous drag in the second mode ofthe present invention.

FIG. 17 shows the relationship between the strain and the viscous dragin the second mode of the present invention.

FIG. 18 shows time history data of a strain ε in a third mode of thepresent invention of the present invention.

FIG. 19 shows time history data of a strain speed ε′ in the third modeof the present invention.

FIG. 20 shows time history data of a stress σ in the third mode of thepresent invention.

FIG. 21 shows a strain-stress curve and a method of computing a modulusof longitudinal elasticity in the third mode of the present invention.

FIG. 22 shows time history data of a viscous drag in the third mode ofthe present invention.

FIG. 23 shows the relationship between the strain and the viscous dragin the third mode of the present invention.

FIG. 24 shows the relationship between the strain and the strain speedin the third mode of the present invention.

FIG. 25 is an illustrative front view showing a split Hopkinson rodtesting machine.

FIG. 26 shows a state of a time history of a strain of a specimen.

BEST MODE FOR CARRYING OUT THE INVENTION

The modes of the present invention will be described below withreference to the drawings.

The mode of the first invention is described below. In the simulationmethod according to the first mode of the present invention, as aviscoelastic material showing nonlinearity, a material containingbutadiene rubber as its main component which is a composing material ofa golf ball is used. By using the above-described material containingthe butadiene rubber as its main component as a specimen and a splitHopkinson rod testing machine improved in its construction, a strain, astrain speed, and a stress generated in the material containing thebutadiene rubber as its main component are measured momently in ameasuring condition in which the material deforms greatly at a highspeed, assuming that a golf ball made of the material is actually used.The measuring method to be carried out by using the split Hopkinson rodtesting machine will be described later.

The split Hopkinson rod testing machine is capable of measuring propertyof the material in various regions of the strain and the strain speed byaltering the collision speed of a hitting rod. In the first mode of thepresent invention, four different patterns of the collision speed (7m/s, 14 m/s, 20 m/s, and 25 m/s) are adopted, and the property of thematerial are measured in four different measuring conditions to obtaintime history data of the strain, the strain speed, and the stress ineach of the four collision patterns. FIGS. 1, 2, and 3 show the timehistory data of the strain ε, the strain speed ε′, and the stress σrespectively.

As shown in FIG. 1, after the hitting rod collides with the material byusing the split Hopkinson rod testing machine, the material containingbutadiene rubber as its main component generates a strain. The value ofthe strain increases in a certain period of time and then decreasesgently. As shown in FIG. 2, the strain speed shows positive values untilthe strain attains a maximum value, and then shows negative values. Asshown in FIG. 3, the value of the stress changes with the elapse oftime.

FIG. 4 shows a strain-stress curve drawn on the basis of the timehistory data of the strain and the stress. In the graph of FIG. 4, themodulus of longitudinal elasticity E of the specimen in each of thecollision patterns is computed by using the maximum strain value ε_(max)and the stress value σ_(a) corresponding to the maximum strain valueε_(max) and by using an equation (4) shown below.

(Equation 4)E=σ _(a)/ε_(max)   (4)

In the case of the material containing the butadiene rubber as its maincomponent used in the first mode of the present invention, even when thecollision speed of the hitting rod of the split Hopkinson rod testingmachine is altered, the specimen has the same modulus of longitudinalelasticity in each collision pattern.

To conduct a simulation in consideration of the viscosity of a productcomposed of the viscoelastic material, a viscoelastic model in which theviscosity of the viscoelastic material is considered is set. Morespecifically, in the first mode, a fundamental two-component Voightmodel shown in FIG. 5 is used as the viscoelastic model consisting of aspring and a dashpot. That is, the viscoelastic material model is usedin such a way that the viscous drag η of the dashpot and the rigidity ofthe spring (modulus of longitudinal elasticity E or shear coefficient)are variable.

As shown in FIG. 5, in the fundamental two-component Voight model usedas the viscoelastic model, assuming that a stress generated in thespring is σ₁ and a stress generated in the dashpot is a σ₂, a stress σgenerated by the entire viscoelastic model can be expressed by anequation (5) shown below:(Equation 5) $\begin{matrix}\begin{matrix}{\sigma = {\sigma_{1} + \sigma_{2}}} \\{= {{E\quad ɛ} + {\eta\quad ɛ^{\prime}}}}\end{matrix} & (5)\end{matrix}$

Therefore from the equation (5), the viscous drag η of the viscoelasticmodel can be expressed by an equation (6) shown below:

(Equation 6)η=(σ−Eε)/ε′  (6)

Therefore from the time history data of the strain ε, the strain speedε′, and the stress a shown in FIGS. 1, 2, and 3 respectively and theequation (3), the viscous drag η of the viscoelastic model correspondingto the strain ε as well as the strain speed ε′ can be computed momently.As described above, the value of each of the strain, the strain speed,and the stress changes with the elapse of time. The state of the straingenerated in the viscoelastic material can be divided into a “strainincrease state” in which the applied strain increases in a compressiondirection thereof and a “strain restoration state” in which acompression amount thereof decreases gradually. Therefore the viscousdrag is computed separately in each of the case in which straingenerated in the viscoelastic material is in the strain increase stateand the case in which strain generated therein is in the strain decreasestate (or strain restoration state) to obtain the time history data ofthe viscous drag at each of the collision speeds, as shown in FIG. 6.

Based on the strain and the stress obtained by the measurement conductedunder the condition of the high speed and the great deformation, themodulus of longitudinal elasticity E of the viscoelastic material isdetermined in consideration of the viscosity thereof. Then by using theequation (6) derived from the viscoelastic model in which the viscosityis taken into consideration, the viscous drag η is computed separatelyin the strain increase state and the strain decrease state (or strainrestoration state). The obtained viscous drag η is applied to thesimulation.

FIG. 7 shows the relationship between the strain and the viscous dragobtained from the time history data of the strain and the viscous drag.As shown in FIG. 7, at an equal value of the strain, the value of theviscous drag in the strain increase state is different from the valuethereof in the strain decrease state (or strain restoration state). FIG.8 shows the relationship between the strain and the strain speedobtained from the time history data of the strain and the strain speed.As shown in FIG. 8, the viscous drag changes in dependence on thevariation of the strain and the strain speed, and the value thereof inthe strain increase state is different from the value thereof in thestrain decrease state (or strain restoration state). Thus the viscousdrag can be determined in dependence on the strain and the strain speedand is computed separately in each of the strain increase state and thestrain decrease state (or strain restoration state). That is, theviscous drag is the function of the strain and the strain speed. In FIG.8, the data at the four different collision speeds look as though theyoverlap each other linearly in the negative region of the strain speed.However the actual values of the strain and those of the strain speed atthe four different collision speeds are different from each otherrespectively.

The simulation is conducted by writing the relationship between thestrain and the viscous drag shown in FIG. 7 and the relationship betweenthe strain and the strain speed shown in FIG. 8 as input data in eachcollision pattern, as will be described later and by making an analysisbased on the finite element method in consideration of a change of theviscous drag that occurs in dependence on a variation of the strain andthe strain speed and in consideration of a change of the viscous dragthat occurs in dependence on the strain increase state and the straindecrease state (or strain restoration state).

In the first mode of the present invention, a golf ball model is set asa product model composed of the viscoelastic material to conduct thesimulation, assuming that a golf club head collides with the golf ball.As shown in FIG. 9, a golf ball model 10 is assumed to be a one-pieceball containing butadiene rubber as its main component and has adiameter of 42.8 mm.

In executing the analysis based on the finite element method, an initialcondition is set on the product model. That is, the initial condition ofthe size, configuration, construction, and the like of the golf ballmodel 10 is set, and the golf ball model 10 is divided into a largenumber of mesh elements 11 to obtain a large number of nodal points 12.The total number of the mesh elements of the golf ball model 10 isfavorably 1000-100000 and more favorably 2500-20000. The upper limitvalue is set in view of the capability of a computer currentlyavailable. As the capability of the computer is improved, a time periodrequired to execute the analysis based on the finite element method isreduced. Thus it is easy to imagine that the upper limit value changes.

Based on the above-described set condition, the data of the golf ballmodel 10 and the relationship among the strain, the strain speed, andthe viscous drag are written as the input data in computations to beperformed in the simulation. When computations are performed, anappropriate viscous drag is computed for each element momently, andcomputations of the equations are performed by using the viscous drags.In the simulation, two-dimensional data of the relationship between thestrain and the strain speed and the relationship between the strain andthe viscous drag in each of the strain increase state and the straindecrease state (or strain restoration state) are inputted to performcomputations. From time series data of the strain and the strain speedin each of the four patterns different from one another in measuringconditions, the correspondence relationship between the strain and thestrain speed in each pattern is recorded, and the strain, the strainspeed, and the viscous drag corresponding to the strain as well as thestrain speed are written as the input data. In the case of a strain anda strain speed not the same as the data of the strains and the strainspeeds measured under the four different measuring conditions, a primaryinterpolation is performed by using a binary value of the viscous dragmeasured in a condition of a proximate strain and a proximate strainspeed.

More specifically, when attention is paid to one element, information ofa strain and a strain speed generated in the element is obtained.Thereafter based on the relationship between the strain and the strainspeed obtained from measurement in each collision case (measurementcase) and with reference to the value of the strain speed when thestrain having an equal value is generated in each collision case, twocollision cases each having the value of the strain speed close to thevalue of the strain speed of the attention-given element are searched.By using the value of each of the strain speed and the viscous drag whenthe values of the strains corresponding to the strain speeds of the twocollision cases are equal to each other, the interpolation is performedto compute an appropriate value of the viscous drag corresponding to thestrain and the strain speed generated in the attention-given element.That is, in the first mode of the present invention, a simple primaryinterpolation is performed. By determining the viscous drag independence on the strain and the strain speed generated in each element,the property of the material in an appropriate condition of the strainand the strain speed is expressed for each element.

The strain and the strain speed of an entire coordinate system generatedin each element are resolved into deviation components and volumecomponents. Then the strain and the strain speed of the deviationcomponents are converted from the entire coordinate system into a mainstrain coordinate system and a main strain speed coordinate systemrespectively. Thereafter by using a converted deviation main strain anda converted deviation main strain speed, a viscous drag is determinedfor each coordinate axis from FIGS. 7 and 8, as described above. Byusing the equations 1, 2, and 3, a stress and a strain of the deviationcomponents in which the viscoelasticity is taken into consideration arecomputed from on the viscous drag. Thereafter the stress of thedeviation components and the strain thereof are re-converted into theentire coordinate system. Based on the stress of the volume componentsand the strain thereof and the stress of the deviation components andthe strain thereof, the stress and strain of each element are computed.

More specifically, six strain components generated in each element areconverted into the main strain coordinate system to determine threestrain components in a main axis direction. The strain and the strainspeed of the entire coordinate system generated in each element areresolved into the volume components and the deviation components inwhich the viscoelasticity is taken into consideration. Six deviationcomponents of the resolved strain and strain speed are converted intothe main strain coordinate system and main strain speed coordinatesystem from the entire coordinate system. Thereafter by using threecomponents of the deviation main strain in the main strain coordinatesystem and three components of the deviation main strain speed in themain strain speed coordinate system, the viscous drag is determined foreach axis of the main strain coordinate system and the main strain speedcoordinate system.

In the first mode of the present invention, as shown in FIGS. 10A, 10B,and 10C, the simulation is conducted by analyzing the property of thematerial for the golf ball model 10 by using the finite element method,when an aluminum cylindrical hollow rod model 20, having a weight of 200g (equal to weight of golf club head), serving as a hitting objectcollides with the golf ball model 10 at a speed of 45 m/s. Therebycomputations is performed to find the amount of the strain generated ineach element 11 of the golf ball model 10 in a predetermined period oftime from the time of collision between the golf ball model 10 and thecylindrical hollow rod model 20.

A mass is distributed to each of nodal points constituting one element.Each nodal point is replaced with a material point. The speed of eachnodal point is regarded as the speed of the material point. Thus thespeeds of total the nodal points are divided by the number of the nodalpoints to obtain the speed of the ball. That is, the speed Vbi (i=x, y,z) of the impacted ball is computed as follows by an equation (7) shownbelow. The entire momentum of the ball is regarded as the sum of themomentums of all the material points. Thus a value obtained by dividingall the momentums by the total weight is defined as the launching speedVbi of the ball.(Equation 7) $\begin{matrix}{{Vbi} = \frac{\sum\limits_{n = 1}^{N}{{Mn}\quad{Vni}}}{M}} & (7)\end{matrix}$

Where N is the number of the nodal points, M is the total mass, Vni isan n-th translation speed, and Mn is a value obtained by dividing themass of an element including the nodal points by the number of the nodalpoints included in the element.

A circular surface 20 a of the cylindrical hollow rod model 20 made ofaluminum is set as a collision surface. The collision surface is flat.The cylindrical hollow rod model 20 collides head-on with the golf ballmodel 10. A central point 20 b of the circular surface 20 a of thecylindrical hollow rod model 20 collides first with the golf ball model10.

By using the above-described method, the speed of the cylindrical hollowrod model 20 and that of the golf ball 10 before and after the collisiontherebetween are computed. The restitution coefficient of the golf ball10 is computed from the speed and weight of each of the cylindricalhollow rod model 20 and the golf ball 10 to estimate the performance ofthe golf ball model 10.

As described above, the strain and the strain speed are resolved intothe deviation components and the volume components. By using the mainstrain and main strain speed of the deviation components resolved intothe main strain coordinate system and the main strain speed coordinatesystem respectively, the viscous drag is determined for the coordinateaxis of each of the main strain coordinate system and the main strainspeed coordinate system in such a way that for each element, the viscousdrag is variable for three components in the direction of each axis.Therefore it is possible to make a correct analysis of athree-dimensional direction in consideration of the viscoelasticity ofthe viscoelastic material and improve the simulation precision.

The simulation is conducted by determining the viscous drag computedseparately in the strain increase state and the strain decrease state(or strain restoration state) as the function of the strain and thestrain speed, inputting the relationship among the strain, the strainspeed, and the viscous drag to the golf ball model, and executing ananalysis based on the finite element method. Therefore it is possible tocompute the viscous drag corresponding to the strain and the strainspeed of each element momently from the strain and the strain speed inconsideration of the difference in the value of the viscous drag betweenthe strain increase state and the strain decrease state (or strainrestoration state) and estimate with very high precision thecharacteristic of the viscoelastic material having a high nonlinearityin the condition of a high speed and a great deformation.

Therefore in a condition equivalent to a condition in which the golfball model is hit with an actual golf club head, it is possible tocorrectly apprehend the performance of the golf ball model such as therestitution coefficient and dynamic behavior of the golf ball modelcomposed of the viscoelastic material without making the golf ball as anexperiment.

From a stress-strain curve of FIG. 11, a phase angle δ is computed byusing an equation (8) shown below. A loss coefficient (tan δ) can becomputed from the phase angle δ.

(Equation 8)δ=sin⁻¹((σ_(a)−σ_(b))/σ_(max))   (8)The mode of the second invention is described below. In the simulationmethod according to the second mode of the present invention, as aviscoelastic material showing nonlinearity, a material containingurethane rubber as its main component which is a composing material of agolf ball is used. By using the above-described material containing theurethane rubber as its main component as a specimen and a splitHopkinson rod testing machine improved in its construction, a strain, astrain speed, and a stress generated in the material containing theurethane rubber as its main component are measured momently in ameasuring condition of a high speed a great deformation, assuming thatthe golf ball is actually used.

In the second mode of the present invention, four different patterns ofthe collision speed (7 m/s, 14 m/s, 20 m/s, and 25 m/s) are adopted, andthe property of the material is measured in four different measuringconditions to obtain time history data of the strain, the strain speed,and the stress in each of the four collision patterns. FIGS. 12, 13, and14 show the time history data of the strain ε, the strain speed ε′, andthe stress σ in each of the four collision patterns respectively.

FIG. 15 shows a strain-stress curve drawn on the basis of the timehistory data of the strain and that of the stress. In the strain-stresscurves of FIG. 15, the maximum strain value ε_(max) and the stress valueσ_(a) corresponding to the maximum strain value ε_(max) are substitutedinto the above-described equation (4) to find the modulus oflongitudinal elasticity E of the specimen in each collision pattern. InFIG. 15, the modulus of longitudinal elasticity E at each collisionspeed is denoted as E1, E2, E3, and E4. In the material containing theurethane rubber as its main component, the value of the modulus oflongitudinal elasticity which is one of rigidities thereof has a change(E1: 9.442 Mpa, E2: 9.839 Mpa, E3: 6.876 Mpa, E4: 6.251 Mpa) in eachcollision pattern, namely, in deformation states of the material.

Similarly to the first mode of the present invention, a viscoelasticmodel consisting of a spring and a dashpot as shown in FIG. 5 is used tofind the viscous drag η from the equations (5) and (6).

Therefore from the time history data of the strain ε, the strain speedε′, and the stress a shown in FIGS. 12, 13, and 14 respectively, themodulus of longitudinal elasticity E corresponding to the strain ε aswell as the strain speed ε′, and the equation (3), it is possible tofind the viscous drag η momently in consideration of the modulus oflongitudinal elasticity E which changes in correspondence to a variationof the strain ε and the strain speed ε′, as shown in FIG. 16.

Based on the strain and the stress obtained by the measurement conductedunder the condition of the high speed and the great deformation, therigidity of the viscoelastic material such as the modulus oflongitudinal elasticity and the modulus of transverse elasticity in eachcondition is determined. Thereafter the relationship among the strain,the strain speed, the rigidity such as the modulus of longitudinalelasticity is adjusted. Then the viscous drag η is computed from theequation (6) derived from the viscoelastic model in which the viscosityis taken into consideration. The obtained viscous drag η is applied tothe simulation.

More specifically, the simulation is conducted by writing therelationship between the strain, the strain speed, and the modulus oflongitudinal elasticity and the relationship between the strain, thestrain speed, and the viscous drag as input data at each collisionspeed, as will be described later and by executing an analysis based onthe finite element method with reference to a change of the rigiditysuch as the modulus of longitudinal elasticity and in consideration of achange of the viscous drag that occurs in dependence on a variation ofthe strain and the strain speed.

Similarly to the first mode of the present invention, in the second modethereof, a one-piece ball containing the urethane rubber as its maincomponent is set as the product model composed of the viscoelasticmaterial to conduct the simulation, assuming that a golf club head(hitting object) collides with a golf ball.

Based on a set condition similar to that of the first mode of thepresent invention, the relationship among the strain, the strain speed,the rigidity such as the modulus of longitudinal elasticity, and theviscous drag and the data of the golf ball model 10 are written as theinput data in executing computations for the simulation. Whencomputations are performed, the rigidity of each element such as themodulus of longitudinal elasticity is computed momently, andcomputations are performed by using the rigidity such as the obtainedmodulus of longitudinal elasticity. In the simulation of the second modeof the present invention, from time series data of the strain and thestrain speed in each of the four patterns different in the measuringcondition, the correspondence relationship among the strain, the strainspeed, and the modulus of longitudinal elasticity as shown in FIG. 17 isrecorded, and the strain, the strain speed, and the modulus oflongitudinal elasticity corresponding to the strain as well as thestrain speed are written as the input data. In the case of a strain anda strain speed not identical to the data of the strains and the strainspeeds measured under the four different measuring conditions, a primaryinterpolation is performed by using a binary value of the modulus oflongitudinal elasticity measured in a condition of a proximate strainand a proximate strain speed.

In FIG. 17, in the negative region of the strain speed ε′, the curvesmeasured in the different conditions look as though they overlap eachother and have an equal value. However the curves have different strainspeeds ε′. Thus the modulus of longitudinal elasticity can be determinedfrom a binary value of the strain and the strain speed.

By determining the modulus of longitudinal elasticity of each element independence on the strain and the strain speed generated therein, theproperty of the material in appropriate strain and strain speedconditions is expressed for each element.

The strain and the strain speed of the entire coordinate systemgenerated in each element are resolved into deviation components andvolume components. As described above, the viscous drag and the rigiditysuch as the modulus of longitudinal elasticity are determined for eachcoordinate axis. Then by using the equations 1, 2, and 3, the stress andthe strain of the deviation components are computed in consideration ofthe viscoelasticity.

The speed of the cylindrical hollow rod model 20 and that of the golfball 10 before and after the collision therebetween are computed byusing a method similar to that of the first mode of the presentinvention. The restitution coefficient of the golf ball 10 is computedfrom the speed and weight of each of the cylindrical hollow rod model 20and the golf ball 10 to estimate the performance of the golf ball 10.

As described above, the strain and the strain speed are resolved intothe deviation components and the volume components. By using the mainstrain and main strain speed of the deviation components resolved intothe main strain coordinate system and the main strain speed coordinatesystem, the viscous drag and the rigidity such as the modulus oflongitudinal elasticity are determined for each axis of the main straincoordinate system and the main strain speed coordinate system in such away that for each element, the viscous drag and the rigidity such as themodulus of longitudinal elasticity are variable for three components inthe direction of each axis. Therefore it is possible to make a correctanalysis in a three-dimensional direction in consideration of theviscoelasticity of the viscoelastic material and improve the simulationprecision. It is also possible to compute a loss coefficient (tan δ).

Therefore in a condition equivalent to a condition in which a golf ballis hit with an actual golf club head, it is possible to correctlyapprehend the performance of the golf ball model composed of theviscoelastic material such as the restitution coefficient and dynamicbehavior without making the golf ball as an experiment.

Similarly to the first mode of the present invention, in the simulationmethod of the third mode of the present invention, the value of each ofa strain, a strain speed, and a stress generated in a materialcontaining butadiene rubber as its main component is measured momently.FIGS. 18, 19, and 20 show time history data of each of the strain ε, thestrain speed ε′, and the stress σ at each collision patternrespectively.

As shown in FIG. 18, when a hitting rod collides with the material byusing the split Hopkinson rod testing machine, the material containingthe butadiene rubber as its main component generates a strain. The valueof the strain increases in a certain period of time and then decreasesgently. As shown in FIG. 19, the strain speed shows positive valuesuntil the strain attains a maximum value, and then shows negativevalues. As shown in FIG. 20, the value of the stress changes with theelapse of time.

FIG. 21 shows a strain-stress curve drawn on the basis of the timehistory data of the strain and the stress. In the strain-stress curve ofFIG. 21, the modulus of longitudinal elasticity E of a specimen iscomputed at each of the collision patterns by using a maximum strainvalue ε_(max) and a stress value σ_(a) corresponding to the maximumstrain value ε_(max) and the above-described equation (4).

Similarly to the first mode of the present invention, in the third modeof the present invention, a viscoelastic model consisting of a springand a dashpot as shown in FIG. 5 is used to find the viscous drag n fromthe equations (2) and (3).

Therefore from the time history data of the strain ε, the strain speedε′, and the stress a shown in FIGS. 18, 19, and 20 respectively and theequation (3), the viscous drag η corresponding to the strain ε as wellas the strain speed ε′ can be computed momently. As described above, thevalue of each of the strain, the strain speed, and the stress changeswith the elapse of time. The state of the strain generated in theviscoelastic material can be divided into a “strain increase state” inwhich the applied strain increases in a compression direction and a“strain restoration state” in which a compression amount decreasesgradually. Therefore the viscous drag is computed separately in the casewhere the stress generated in the viscoelastic material is in the strainincrease state and the case where the stress generated therein is in thestrain decrease state (or strain restoration state) to obtain the timehistory data of the viscous drag at each of the collision speeds, asshown in FIG. 22.

Based on the strain and the stress obtained by the measurement conductedunder the condition of a high speed and a great deformation, the modulusof longitudinal elasticity E of the viscoelastic material is determined.Then by using the equation (6) derived from the viscoelastic model inwhich the viscosity is taken into consideration, the viscous drag η ofthe viscoelastic material is computed separately in the strain increasestate and the strain decrease state (or strain restoration state). Theviscous drag η different between the strain increase state and in thestrain decrease state are applied to the simulation.

FIG. 23 shows the relationship between the strain and the viscous dragobtained from the time history data of the strain and that of theviscous drag. As shown in FIG. 23, at an equal value of the strain, thevalue of the viscous drag in the strain increase state is different fromthe value thereof in the strain decrease state (or strain restorationstate). FIG. 24 shows the relationship between the strain and the strainspeed obtained from the time history data of the strain and that of thestrain speed. As shown in FIG. 24, the viscous drag changes independence on variations of the strain and the strain speed and isdifferent between the strain increase state and the strain decreasestate (or strain restoration state). Thus the viscous drag can bedetermined in dependence on the strain and the strain speed. The viscousdrag is the function of the strain and the strain speed in each of thestrain increase state and the strain decrease state. In FIG. 24, thedata of the four different collision speeds look as though they overlapeach other linearly in the negative region of the strain speed. Howeverthe actual values of the strain and those of the strain speed at thefour different collision speeds are different from each otherrespectively.

The simulation is conducted by writing the relationship between thestrain and the viscous drag shown in FIG. 23 and the relationshipbetween the strain and the strain speed shown in FIG. 24 as input dataat each collision speed, as will be described later and by executing ananalysis based on the finite element method in consideration of avariation of the viscous drag in dependence on the difference in thestrain and the strain speed, namely, the difference in the viscous dragbetween the strain increase state and the strain decrease state (orstrain restoration state).

Similarly to the first mode of the present invention, in the third modeof the present invention, a one-piece ball containing the butadienerubber as its main component is set as a product model composed of theviscoelastic material to conduct a simulation, assuming that a golf clubhead (hitting object) collides with the golf ball.

Based on a set condition similar to that of the first mode of thepresent invention, the data of the golf ball model 10 and therelationship among the strain, the strain speed, and the viscous dragare written as the input data in computations to be performed for thesimulation. When computations are executed, an appropriate viscous dragis computed for each element momently from the equations, andcomputations of the equations are performed by using the viscous drags.In the simulation, two-dimensional data of the relationship between thestrain and the strain speed and the relationship between the strain andthe viscous drag in the strain increase state and the strain decreasestate (or strain restoration state) are separately inputted to performcomputations. From time series data of the strain and the strain speedin each of the four patterns different in the measuring condition, thecorrespondence relationship between the strain and the strain speed isrecorded, and the strain, the strain speed, and the viscous dragcorresponding to the strain as well as the strain speed are written asthe input data. In the case of a strain and a strain speed not identicalto the data of the strains and the strain speeds measured under the fourdifferent measuring conditions, a primary interpolation is performed byusing a binary value of the modulus of longitudinal elasticity measuredin a condition of a proximate strain and a proximate strain speed.

More specifically, when attention is paid to one element, the magnitude(norm) of the element is computed, and a computed norm is compared witha norm of a previous step in the simulation. If there is an increase inthe norm, it is determined that the strain is in the increase state. Ifthere is a decrease in the norm, it is determined that the strain is inthe decrease state. By determining on whether the strain is in theincrease state or the decrease state, the information of the strain andthe strain speed is obtained in consideration of the state of thematerial. Thereafter based on the relationship between the strain andthe strain speed obtained from measurement in each collision case(measurement case) and with reference to the value of the strain speedwhen the strain having an equal value is generated in each collisioncase, two collision cases each having the value of the strain speedclose to the value of the strain speed of the attention-given elementare searched. By using the value of each of the strain speed and theviscous drag when the values of the strains corresponding to the strainspeeds of the two collision cases are equal to each other, theinterpolation is performed to compute an appropriate value of theviscous drag corresponding to the strain and the strain speed generatedin the attention-given element. That is, in the third mode of thepresent invention, the simple primary interpolation is performed. Bydetermining different viscous drags of each element in the strainincrease state and the strain decrease state in dependence on the strainand the strain speed generated therein, the property of the material inan appropriate condition of the strain and the strain speed is expressedfor each element.

The strain and the strain speed of the entire coordinate systemgenerated in each element are resolved into deviation components andvolume components. As described above, for strains having an equalvalue, the viscous drag different between the strain increase state andthe strain decrease state or the strain restoration state is determinedfor each coordinate axis. Then from the equations 1, 2, and 3, thestress and the strain of the deviation components are computed inconsideration of the viscous drag based on the viscous drag.

The speed of the cylindrical hollow rod model 20 and that of the golfball 10 before and after the collision therebetween are computed byusing a method similar to that of the first mode of the presentinvention. The restitution coefficient of the golf ball 10 is computedfrom the speed and weight of each of the cylindrical hollow rod model 20and the golf ball 10 to estimate the performance of the golf ball 10.

As described above, the strain and the strain speed are resolved intothe deviation components and the volume components. By using the mainstrain and main strain speed of the deviation components resolved intothe main strain coordinate system and the main strain speed coordinatesystem respectively, for each axis of the main strain coordinate systemand the main strain speed coordinate system, a viscous drag differentbetween the strain increase state and the strain decrease or the strainrestoration state is determined for the strain having an equal value insuch a way that for each element, the viscous drag is variable for threecomponents in the direction of each axis. Therefore it is possible tomake a correct analysis in a three-dimensional direction inconsideration of the viscoelasticity of the viscoelastic material andimprove the simulation precision. It is also possible to compute a losscoefficient (tan δ).

The simulation is conducted by determining the viscous drag computedseparately in the strain increase state and the strain decrease state(or strain restoration state) as the function of the strain and thestrain speed, inputting the relationship among the strain, the strainspeed, and the viscous drag to the golf ball model, and executing ananalysis based on the finite element method. Therefore it is possible tocompute the viscous drag corresponding to the strain and the strainspeed of each element momently from the strain and the strain speed inconsideration of the difference in the viscous drag between the strainincrease state and the strain decrease state (or strain restorationstate) and estimate with very high precision the characteristic of theviscoelastic material having a high nonlinearity in the condition of ahigh speed and a great deformation.

Therefore in a condition equivalent to a condition in which a golf ballis hit with an actual golf club head, it is possible to correctlyapprehend the performance the golf ball model composed of theviscoelastic material such as the restitution coefficient and dynamicbehavior thereof without making the golf ball as an experiment.

Measurement Conducted by Split Hopkinson Rod Testing Machine to MeasureProperty of Material

FIG. 25 is an illustrative front view showing a split Hopkinson rodtesting machine improved in its construction to measure the viscoelasticmaterial.

The split Hopkinson rod testing machine shown in FIG. 25 has a hittingrod 51, an input rod 53, and an output rod 55. These rods are arrangedlinearly. A first strain gauge 57 and a second strain gauge 59 areinstalled on the input rod 53. A third strain gauge 61 and a fourthstrain gauge 63 are installed on the output rod 55. A columnar specimen70 is sandwiched between a rear end 53 a of the input rod 53 and a frontend 55 a of the output rod 55.

The specimen 70 may be formed by molding the viscoelastic material intoa predetermined configuration or cutting a product composed of theviscoelastic material by molding the viscoelastic material into thepredetermined configuration. In the mode of the present invention, thespecimen 70 has a length (distance between the rear end 53 a of theinput rod 53 and the front end 55 a of the output rod 55) of 4 mm and asectional diameter of 18 mm.

The hitting rod 51, the input rod 53, and the output rod 55 arecylindrical and made of polymethyl methacrylate. The sectional diameterof each of the hitting rod 51, the input rod 53 and the output rod 55 isset to 20 mm. The modulus of longitudinal elasticity of each of thehitting rod 51, the input rod 53 and the output rod 55 is set to 5300Mpa. The specific gravity of each of the hitting rod 51, the input rod53 and the output rod 55 is set to 1.19. The length of the hitting rod51 is set to 100 mm. The length of each of the input rod 53 and theoutput rod 55 (hereinafter may be referred to as stress rod) is set to2000 mm.

The first strain gauge 57 is installed on the input rod 53 at a positionspaced 900 mm from the rear end 53 a thereof. The second strain gauge 59is installed on the input rod 53 at a position spaced 600 mm from therear end 53 a thereof. The third strain gauge 61 is installed on theoutput rod 55 at a position spaced 300 mm from the front end 55 athereof. The fourth strain gauge 63 is installed on the output rod 55 ata position spaced 600 mm from the front end 55 a thereof.

In the split Hopkinson rod testing machine shown in FIG. 25, the hittingrod 51, the input rod 53, and the output rod 55 are made of syntheticresin consisting of polymethyl methacrylate. The input rod 53 and theoutput rod 55 are as long as 2000 mm. The distance between the firststrain gauge 57 and the rear end 53 a of the input rod 53 is long. Thedistance between the second strain gauge 59 and the rear end 53 a of theinput rod 53 is also long. Therefore, the split Hopkinson rod testingmachine is suitable for measuring the strain, the strain speed, and thestress of a viscoelastic material such as crosslinked rubber which isused for a golf ball.

A monoaxial strain gauge for plastic is used as the first strain gauge57, the second strain gauge 59, the third strain gauge 61, and thefourth strain gauge 63. In the mode of the present invention, amonoaxial strain gauge KFP-5-350-C1-65 manufactured by Kyowa DengyoKabushiki Kaisha is used. The monoaxial strain gauge is bonded to theabove-described positions of the input rod 53 and the output rod 55. Thefirst strain gauge 57 through the fourth strain gauge 63 are installedon the input rod 53 and the output rod 55 linearly in the longitudinaldirection thereof.

In measuring the strain of the specimen, its strain speed, and itsstress with the split Hopkinson rod testing machine, initially, thehitting rod 51 is brought into collision with the front end 53 b of theinput rod 53 at a predetermined speed. Thereby, an incident distortedwave is generated in the input rod 53. The incident distorted waveadvances to the rear end 53 a of the input rod 53. A part of theincident distorted wave is reflected from the rear end 53 a of the inputrod 53 to generate a reflected distorted wave. The reflected distortedwave advance to the front end 53 b of the input rod 53. A part of theincident distorted wave advances to the specimen 70 from the rear end 53a of the input rod 53 and propagates to the output rod 55 to generate atransmitted distorted wave. The transmitted distorted wave advances tothe rear end 55 b of the output rod 55.

The incident distorted wave is measured with the first strain gauge 57and the second strain gauge 59. The incident distorted wave is passedthrough a low-pass filter to remove a high-frequency wave having afrequency more than 10 KHz from the incident distorted wave. Zerocompensation is performed to make the base line value of the timehistory of the incident distorted wave zero. Fourier transformation ofan obtained time base strain at each of the first strain gauge 57 andthe second strain gauge 59 is performed to determine a frequency axisstrain. A transmission function is derived from the frequency axisstrain at the first strain gauge 57 and the second strain gauge 59.Based on the transmission function, the frequency axis strain at therear end 53 a of the input rod 53 is estimated in consideration of theratio of the distance X1 between the first strain gauge 57 and the rearend 53 a of the input rod 53 to the distance X2 between the secondstrain gauge 59 and the rear end 53 a of the input rod 53. Fourierinverse transformation of the frequency axis strain is performed toobtain a time base strain (time history of strain) ε i of the incidentdistorted wave at the rear end 53 a of the input rod 53.

Similarly, the second strain gauge 59 and the first strain gauge 57measure the reflected distorted wave reflected from the rear end 53 a ofthe input rod 53 and advancing to the front end 53 b of the input rod53. A time base strain (time history of strain) ε r of the reflecteddistorted wave at the rear end 53 a of the input rod 53 is obtained fromthe measured reflected distorted wave.

The transmitted distorted wave which propagates to the output rod 55through the specimen 70 is measured with the third strain gauge 61 andthe fourth strain gauge 63 installed on the output rod 55. A time basestrain (time history of strain) ε t of the transmitted distorted wave atthe front end 55 a of the output rod 55 is obtained from the measuredtransmitted distorted wave.

From the obtained time base strains ε i, ε r, and ε t, a strain speed ε′of the specimen 70 is computed by using an equation (9) shown below.(Equation 9) $\begin{matrix}\begin{matrix}{ɛ^{\prime} = {\left( {C_{0}/L} \right) \cdot \left( {{ɛ\quad i} - {ɛ\quad r} - {ɛ\quad t}} \right)}} \\{= {\left( {\left( {E/\rho} \right)^{1/2}/L} \right) \cdot \left( {{ɛ\quad i} - {ɛ\quad r} - {ɛ\quad t}} \right)}}\end{matrix} & (9)\end{matrix}$

(In the equation (9), C₀ indicates the propagation speed (m/s) of thestrain wave in the stress rod and the output rod, L indicates the length(m) of the specimen, E is the modulus of longitudinal elasticity (N/m²)of the stress rod, and ρ is the density (kg/m³) of the stress rod).

From the time base strains ε i, ε r, ε t, the strain ε of the specimen70 is computed by using an equation (10) shown below.(Equation 10) $\begin{matrix}\begin{matrix}{ɛ = {\left( {{CO}/L} \right) \cdot {\int_{0}^{t}{\left( {{ɛ\quad i} - {ɛ\quad r} - {\overset{.}{ɛ}t}} \right){\mathbb{d}t}}}}} \\{= {\left( {\left( {E/\rho} \right)^{1/2}/L} \right) \cdot {\int_{0}^{t}{\left( {{ɛ\quad i} - {ɛ\quad r} - {ɛ\quad t}} \right){\mathbb{d}t}}}}}\end{matrix} & (10)\end{matrix}$

(In the equation (10), C₀ indicates the propagation speed (m/s) of thestrain wave in the stress rod and the output rod, L indicates the length(m) of the specimen, E is the modulus of longitudinal elasticity (N/m²)of the stress rod, and ρ is the density (kg/m³) of the stress rod).

From the time base strains ε i, ε r, and ε t, the stress σ of thespecimen 70 is computed by using an equation (11) shown below.(Equation 11) $\begin{matrix}\begin{matrix}{\sigma = {\left( {E \cdot {A/\left( {2{As}} \right)}} \right) \cdot \left( {{ɛ\quad i} + {ɛ\quad r} + {ɛ\quad t}} \right)}} \\{= {\left( {E \cdot {D^{2}/\left( {2({Ds})^{2}} \right)}} \right) \cdot \left( {{ɛ\quad i} + {ɛ\quad r} + {ɛ\quad t}} \right)}}\end{matrix} & (11)\end{matrix}$

(In the equation 11, E indicates the modulus of longitudinal elasticity(N/m²) of the stress rod consisting of the input rod and the output rod;A indicates the sectional area (m²) of the stress rod; As indicates thesectional area (m²) of the specimen; D indicates the diameter (m) of thestress rod; and Ds indicates the diameter (m) of the specimen.

FIG. 26 shows the obtained strain time history of the specimen 70. Asshown in FIG. 26, the curve is smooth for a certain period of time aftera time corresponding to a peak P. After a time corresponding to a givenpoint of the graph FIG. 26, the curve becomes irregular. A point S isselected in the curve-smooth stage between the peak P and the givenpoint. A tangent to the curve at the point S is drawn. A relaxation timeλ is derived from the intersection of the tangent and the time base. Acurve found by using an equation (12) shown below is determined as thecurve after the point S of FIG. 26. In this manner, the entire straintime history is corrected to a smooth curve (shown with a one-dot linein FIG. 26). Thereby, it is possible to remove the influence of noise onan obtained viscoelastic characteristic value.

(Equation 12)ε(t)=ε₀ ·e ^(−t/λtm ()12)

(In the equation 12, ε₀ is a strain at the point of contact.)

Similarly, it is possible to make an entire stress time history a smoothcurve by using an equation (13) shown below. Thereby, it is possible toremove the influence of noise on an obtained viscoelastic characteristicvalue.

(Equation 13)σ(t)=σ₀ ·e ^(−t/λ)  (13)

(In the equation 13, σ₀ is a stress at the point of contact.)

As described above, the time history of the strain and the stress of thespecimen 70 are corrected.

As the above-described method, by using the split Hopkinson rod testingmachine, the time history data of the strain, the strain speed, and thestress at the time the high speed and the great deformation areobtained.

Examples of experiments of the present invention will be described belowin detail.

(Experiment 1) Using a material containing urethane rubber as its maincomponent, a golf ball was prepared. The material was compression-moldedin a die having a diameter of 42.8 mm at 160° C. for 30 minutes to forma one-piece golf ball.

The strain, strain speed, and stress of the material containing theurethane rubber as its main component were measured with the splitHopkinson rod testing machine in collision speeds (7 m/s, 14 m/s, 20m/s, and 25 m/s) of the hitting rod at a room temperature of 23° C. anda relative humidity of 50%.

The maximum strain and the maximum strain speed in the measurement areshown below.

-   Collision speed 7 m/s (maximum strain: 0.12, maximum strain speed:    1378/s)-   Collision speed 14 m/s (maximum strain: 0.24, maximum strain speed:    2703/s)-   Collision speed 20 m/s (maximum strain: 0.35, maximum strain speed:    3898/s)-   Collision speed 25 m/s (maximum strain: 0.43, maximum strain speed:    4716/s)

Since measurement is made at the four-pattern collision speeds, thephase angle δ at each of the four collision speeds is shown in table 1below. TABLE 1 Collision speed of hitting rod (m/s) 7 14 20 25 Phaseangle δ (rad) in 0.39 0.63 1.02 1.11 experimental result Phase angle δ(rad) of 0.36 0.58 0.98 0.99 experiment example 1 Phase angle δ (rad) of0.77 0.81 0.87 0.87 experiment example 2

EXPERIMENT EXAMPLE 1

A simulation was conducted by using the time history data of each of thestrain, strain speed, and stress of a specimen, containing urethanerubber as its main component, measured by the split Hopkinson rodtesting machine and using a viscoelastic model similar to that of thefirst mode of the present invention in consideration of the viscousdrag. The relationship among the strain, the strain speed, and themodulus of longitudinal elasticity was inputted to the product modelused in the simulation method of the first mode of the present inventionin consideration of a change of the modulus of longitudinal elasticitywhich occurs in dependence on a variation of the strain and the strainspeed. Further the viscous drag was considered by differentiating thevalue of the viscous drag between the case where the stress generated inthe viscoelastic material is in the strain increase state and the casewhere the stress generated therein is in the strain decrease state. Asoftware used was a general-purpose software Ls-dyna 940 manufactured byNippon Sogo Kenkyusho Kabushiki Kaisha.

A phase angle δ estimated by the simulation of the experiment example 1is shown in table 1.

EXPERIMENT EXAMPLE 2

With reference to the loss coefficient of a specimen composed of aviscoelastic material containing urethane rubber as its main componentmeasured by the split Hopkinson rod testing machine, a simulation wasconducted on the conventional viscoelastic material whose viscous dragdoes not change.

A phase angle δ estimated by the simulation of the experiment example 2is shown in table 1.

By performing an analysis based on the finite element method, analuminum hollow rod model having a weight of 200 g (equal to weight ofgolf club head) collided with a golf ball made of the materialcontaining the urethane rubber as its main component at speeds of 35m/s, 40 m/s, and 45 m/s to simulate the performance of the golf ball andthe deformation state of the material. By performing an analysis basedon the finite element method, the restitution coefficient of the golfball was computed. Table 2 shown below indicates the restitutioncoefficient of the golf ball, made of the material containing urethaneas its main component, obtained by performing an analysis based on thefinite element method in the experiment examples 1 and 2. TABLE 2Initial speed (m/s) of aluminum hollow rod (m/s) 35 40 45 Restitution0.63 0.60 0.56 coefficient (experiment) Restitution 0.46 0.44 0.42coefficient (experiment example 2) Difference from experiment (%) −27.02−25.80 −24.70 Restitution 0.60 0.56 0.53 coefficient (experimentexample 1) Difference from experiment (%) −4.82 −5.26 −5.98

An experiment was conducted by using an actual golf ball made by moldingthe material containing urethane as its main component to measure therestitution coefficient of the golf ball by the following method.

Table 2 also shows the difference (%) between the restitutioncoefficient obtained in the experiment using the following actual golfball and the restitution coefficients obtained by the analysis made inthe experiment examples 1 and 2.

Measurement of Restitution Coefficient in Experiment Conducted by UsingActual Golf Ball

As the method of measuring the restitution coefficient of the golf ball,an aluminum hollow rod serving as a substitution of a golf club head andhaving a weight of 200 g (equal to weight of golf club head) collidedwith a golf ball made of the above-described material at speeds of 35,40, and 45 m/s at a temperature of 23° C. The speed of the hollow rodand that of the golf ball before and after the collision therebetweenwere measured. The restitution coefficient of the golf ball was computedfrom the speed and weight of the hollow rod and the golf ball.

The collision surface of the hollow rod was flat. The hollow rodcollided head-on with the golf ball. Since the hollow rod was notcornered like the golf club head, the golf ball did not rotate when bothcollided with each other. Thus only the restitution coefficient of thegolf ball could be evaluated.

As shown in table 1, the phase angle δ in the analysis of the experimentexample 1 was almost equal to the phase angle δ in the experiment ateach collision speed. On the other hand, the phase angle δ in theanalysis of the experiment example 2 had a large difference from thephase angle δ in the experiment at each collision speed. It could beconfirmed that the experiment example 1 could simulate the result of theexperiment with high accuracy.

As shown in table 2, at each of the collision speeds of the hollow rodmodel, the value of the restitution coefficient in the analysis of theexperiment example 1 was close to that computed on the actual golf ballprepared in the experiment. The difference in the restitutioncoefficient of the experiment example 1 from that of the experimentalresult of the actual golf ball was −4.82% to −5.98%. This indicates thatthe experiment example 1 could accurately simulate the experimentalresult of the actual golf ball. On the other hand, at each collisionspeed of the hollow rod, the value of the restitution coefficient in theanalysis of the experiment example 2 was much different from thatcomputed from the actual golf ball prepared in the experiment. Thedifference in the restitution coefficient of the experiment example 2from that of experimental result of the actual golf ball was −24.70% to−27.02%. The simulation of the experiment example 2 was much differentfrom the experimental result.

The above-described result indicates that by conducting the simulationin consideration of the viscous drag, it is possible to accuratelyestimate the performance of a product composed of the viscoelasticmaterial showing nonlinearity in a condition equivalent to a state inwhich the product is actually used.

(Experiment 2)

Similarly to the experiment 1, the value of the phase angle δ at each ofthe four collision speeds was computed for a material containingurethane rubber as its main component. The results are shown in table 3below. TABLE 3 Collision speed of hitting rod (m/s) 7 14 20 25 Phaseangle δ (rad) in 0.39 0.63 1.02 1.11 experimental result Phase angle δ(rad) of 0.36 0.58 0.98 0.99 experiment example 3 Phase angle δ (rad) of0.45 0.77 1.06 0.99 experiment example 4

EXPERIMENT EXAMPLE 3

The strain, strain speed, and stress of a specimen containing urethanerubber as its main component were measured by the split Hopkinson rodtesting machine. A simulation was conducted by using the time historydata of each of the measured strain, strain speed, and stress of thematerial and a viscoelastic model similar to that of the first mode ofthe present invention in consideration of a change of the viscous dragand the rigidity such as the modulus of longitudinal elasticity. Therelationship among the strain, the strain speed, and the rigidity suchas the modulus of longitudinal elasticity was inputted to the productmodel used in the simulation method of the first mode of the presentinvention in consideration of a change of the rigidity such as themodulus of longitudinal elasticity which occurs in dependence on avariation of the strain and the strain speed. Further the viscous dragwas considered by differentiating the value of the viscous drag betweenthe case where the stress generated in the viscoelastic material is inthe strain increase state and the case where the stress generatedtherein is in the strain decrease state.

The phase angle δ estimated by the simulation of the experiment example3 is shown in table 3.

EXPERIMENT EXAMPLE 4

Time history data of each of the strain, strain speed, and stress of aspecimen containing urethane rubber as its main component measured bythe split Hopkinson rod testing machine was used. A simulation wasconducted by using the measured strain, strain speed, and stress, withthe rigidity constant at 25 m/s. The experiment example 4 was similar tothe experiment example 3 except the above-described points.

The phase angle δ estimated by the simulation of the experiment example4 is shown in table 3.

Similarly to the experiment 1, the restitution coefficient of a golfball was computed by making an analysis thereof. Table 4 shown belowindicates the restitution coefficient of the golf ball, made of thematerial containing urethane as its main component, obtained byperforming analyses in the experiment examples 3 and 4. TABLE 4 Initialspeed (m/s) of aluminum hollow rod (m/s) 35 40 45 Restitution 0.63 0.60.56 coefficient (experiment) Restitution coefficient (experiment 0.530.48 0.41 example 4) Difference from experiment (%) −15.8 −20 −19.6Restitution coefficient (experiment 0.6 0.56 0.53 example 3) Differencefrom experiment (%) −4.82 −5.26 −5.98

Similarly to the experiment 1, the value of the restitution coefficientof an actual golf ball was measured. Table 4 shows the difference (%)between the restitution coefficient obtained by the analysis in theexperiment examples 3 and 4 and the restitution coefficient obtained inthe experiment in which the actual golf ball was used.

As shown in table 3, the phase angle δ obtained by the analysis of theexperiment example 3 was almost equal to the phase angle δ obtained inthe experiment. On the other hand, the phase angle δ obtained by theanalysis of the experiment example 4 had a difference from the phaseangle δ obtained in the experiment at each collision speed. Since therigidity was determined at 25 m/s, the rigidity was lower than theactual value. Therefore it could be confirmed that the experimentexample 3 simulated the experimental result with high accuracy. Morespecifically, at the collision speeds of 7, 14, and 20 m/s, therestitution coefficient was computed at a lower rigidity than the actualvalue in the experiment example 4, and the viscous drag was not changed.Thus the phase angle deviated in a larger direction.

As shown in table 4, at each of the collision speeds of the hollow rodmodel, the value of the restitution coefficient in the analysis of theexperiment example 3 was close to that computed on the actual golf ballprepared in the experiment. The difference in the restitutioncoefficient of the experiment example 3 from that of the experimentalresult of the actual golf ball was −4.82% to −5.98%. This indicates thatthe experiment example 3 could accurately simulate the result of theactual golf ball. On the other hand, at each collision speed of thehollow rod, the value of the restitution coefficient in the analysis ofthe experiment example 4 was much different from that computed from theactual golf ball prepared in the experiment. The difference in therestitution coefficient of the experiment example 4 from that ofexperimental result of the actual golf ball was −15.80% to 20.0%.Compared with the experiment example 3, the simulation of the experimentexample 4 was much different from the experimental result.

The above-described result indicates that by conducting the simulationin consideration of the viscous drag and the rigidity such as themodulus of longitudinal elasticity, it is possible to accuratelyestimate the performance of the product composed of the viscoelasticmaterial which shows nonlinearity and in particular, the productcomposed of the viscoelastic material whose rigidity such as the modulusof longitudinal elasticity changes in dependence on a deformation state(magnitude of strain and strain speed) of the material in a conditionequivalent to a state in which the product is actually used.

(Experiment 3)

Similarly to the experiment 1, the value of the phase angle δ at each ofthe four collision speeds was computed for a material containingurethane rubber as its main component. The results are shown in table 5below. TABLE 5 Collision speed of hitting rod (m/s) 7 14 20 25 Phaseangle δ (rad) in 0.39 0.63 1.02 1.11 experimental result Phase angle δ(rad) of 0.36 0.58 0.98 0.99 experiment example 5 Phase angle δ (rad) of0.32 0.53 0.79 0.85 experiment example 6

EXPERIMENT EXAMPLE 5

A simulation was conducted by using time history data of each of thestrain, strain speed, and stress of a specimen, containing urethanerubber as its main component, measured by the split Hopkinson rodtesting machine and using a viscoelastic model similar to that of thethird mode of the present invention in consideration of the viscous dragdifferent between the strain increase state and the strain decreasestate or the strain restoration state. In the simulation method of thethird mode of the present invention, the relationship among the strain,the strain speed, and the rigidity such as the modulus of longitudinalelasticity was inputted to the product model in consideration of achange of the modulus of longitudinal elasticity which occurs independence on a variation of the strain and the strain speed. A softwareused was a general-purpose software Ls-dyna 940 manufactured by NipponSogo Kenkyusho Kabushiki Kaisha.

The phase angle δ estimated by the simulation of the experiment example5 is shown in table 5.

EXPERIMENT EXAMPLE 6

Time history data of each of the strain, strain speed, and stress of aspecimen containing urethane rubber as its main component measured bythe split Hopkinson rod testing machine was used. A simulation wasconducted by making the modulus of longitudinal elasticity variable andutilizing the strain and the strain speed and the viscous drag in thestrain increase state. The experiment example 6 was similar to theexperiment example 5 except the above-described points.

The phase angle δ estimated by the simulation of the experiment exampleδ is shown in above table 5.

Similarly to the experiment example 1, the restitution coefficient of agolf ball was computed by making an analysis thereof. Table 6 shownbelow indicates the restitution coefficient of the golf ball, made ofthe material containing urethane as its main component, obtained byperforming analyses in the experiment examples 5 and 6. TABLE 6 Initialspeed (m/s) of aluminum hollow rod (m/s) 35 40 45 Restitution 0.63 0.60.56 coefficient (experiment) Restitution coefficient (experiment 0.770.71 0.65 example 6) Difference from experiment (%) 22.2 18.3 16.1Restitution coefficient (experiment 0.6 0.56 0.53 example 5) Differencefrom experiment (%) −4.82 −5.26 −5.98

Similarly to the experiment 1, the value of the restitution coefficientof an actual golf ball was measured. Table 6 shows the difference (%)between the restitution coefficient obtained by analyses in theexperiment examples 5 and 6 and the restitution coefficient obtained inan experiment in which the actual golf ball was used.

As shown in table 5, the phase angle δ obtained by the analysis of theexperiment example 5 was almost equal to the phase angle δ obtained inthe experiment at each collision speed. On the other hand, becausesimulation of the experiment example 6 was conducted in the strainincrease state and by using a curve of the strain increase state, theenergy loss was lower than that of the experiment example 5 in thestress-strain curve. Thus the phase angle δ obtained by the analysis ofthe experiment example 6 was smaller than that of the experiment example5 at each collision speed. The phase angle δ obtained by the analysis ofthe experiment example 6 was also smaller than that of the experimentalresult and thus had a large difference from that of the experimentalresult. Therefore it could be confirmed that the experiment example 5simulated the experimental result with high accuracy.

As shown in table 6, at each of the collision speeds of the hollow rodmodel, the value of the restitution coefficient in the analysis of theexperiment example 5 was close to that computed on the actual golf ballprepared in the experiment. The difference in the restitutioncoefficient of the experiment example 5 from that of the experimentalresult of the actual golf ball was −4.82% to −5.98%. This indicates thatthe experiment example 5 could accurately simulate the result of theactual golf ball. On the other hand, the value of the restitutioncoefficient in the analysis of the experiment example 6 was greater thanthat of the experiment since the value of the phase angle δ was smalland had a large difference from that of the experiment example 5. Thedifference in the restitution coefficient of the experiment example 6from that of experimental result of the actual golf ball was 16.1% to22.2%.

As apparent from the foregoing description, it could be confirmed thatsince the simulation is conducted in both the strain increase state andthe strain decrease or the strain restoration state in consideration ofthe difference in the viscous drag between the strain increase state andthe strain decrease or the strain restoration state, it is possible toaccurately estimate the performance of the product composed of theviscoelastic material showing nonlinearity in the condition equivalentto the state in which the product is actually used.

Industrial Applicability

As apparent from the foregoing description, according to the firstinvention, to consider the viscoelastic characteristic in the deviationcomponents of the strain and the strain speed generated in each element,the strain and the strain speed are resolved into the deviationcomponents and the volume components. Then by using the main strain andthe main strain speed of the deviation components converted into themain strain coordinate system and the main strain speed coordinatesystem, the viscous drag is determined for the coordinate axis of eachof the main strain coordinate system and the main strain speedcoordinate system in such a way that for each element, the viscous dragis variable for three components in the direction of each axis.Therefore it is possible to make a correct analysis of athree-dimensional direction in consideration of the viscoelasticity ofthe viscoelastic material and improve the simulation precision.

Further, using the viscoelastic model in which the viscosity of theviscoelastic material is considered, the simulation is conducted byderiving the viscous drag of the viscoelastic material and inputting therelationship among the strain, the strain speed, and the viscous drag tothe product model. Therefore it is possible to accurately express aphenomenon that the property of the viscoelastic material changesnonlinearly in dependence on its deformation speed and magnitude.

In the second invention, to consider the viscoelastic characteristic inthe deviation components of the strain and the strain speed generated ineach element, the strain and the strain speed are resolved into thedeviation components and the volume components. Then by using the mainstrain and the main strain speed of the deviation components convertedinto the main strain coordinate system and the main strain speedcoordinate system respectively, the viscous drag is determined for thecoordinate axis of each of the main strain coordinate system and themain strain speed coordinate system in such a way that for each element,the viscous drag is variable for three components in the direction ofeach axis. Therefore it is possible to make a correct analysis of athree-dimensional direction in consideration of the viscoelasticity ofthe viscoelastic material and improve the simulation precision.

Further the rigidity such as the modulus of longitudinal elasticity ofthe viscoelastic material which changes in dependence on a variation ofthe state of the material is computed from the time history data of thestrain and the stress obtained from measurement to conduct a simulationin consideration of the change of the rigidity such as the modulus oflongitudinal elasticity corresponding to a variation of the strain andthe strain speed. Therefore it is possible to accurately express thephenomenon that the property of the viscoelastic material changesnonlinearly in dependence on its deformation speed and magnitude.

In the third invention, to consider the viscoelastic characteristic inthe deviation components of the strain and the strain speed generated ineach element, the strain and the strain are resolved into the deviationcomponents and the volume components. Then by using the main strain andthe main strain speed of the deviation components converted into themain strain coordinate system and the main strain speed coordinatesystem respectively, the viscous drag is determined for the coordinateaxis of each of the main strain coordinate system and the main strainspeed coordinate system in such a way that for each element, the viscousdrag is variable for three components in the direction of each axis.Therefore it is possible to make a correct analysis of athree-dimensional direction in consideration of the viscoelasticity ofthe viscoelastic material and improve the simulation precision.

Using the viscoelastic model in which the viscosity of the viscoelasticmaterial is considered, the simulation is conducted by deriving theviscous drag of the viscoelastic material separately in the strainincrease state and the strain decrease state or the strain restorationstate and inputting the relationship among the strain, the strain speed,and the viscous drag different between the strain increase state and thestrain decrease state or the strain restoration state to the productmodel. Therefore it is possible to accurately grasp the actual state ofthe material and express the phenomenon that the property of theviscoelastic material changes nonlinearly in dependence on itsdeformation speed and magnitude.

Further according to the first, second, and third inventions, becausethe value of each of the strain, the strain speed, and the stress ismeasured in a measuring condition equivalent to a state in which theproduct composed of the viscoelastic material is actually used, it ispossible to conduct the simulation in correspondence to variousdeformation states of the viscoelastic material.

Accordingly it is possible to accurately analyze the performance and thedynamic behavior of the product composed of the viscoelastic material inwhich the relationship between the strain and the strain speed changesin dependence on a deformation state and whose property such as a losscoefficient shows nonlinearity.

The simulation method is capable of accurately estimating an actualhitting test in the condition of the strain and the strain speedequivalent to those generated in the viscoelastic material of an actualgolf ball when it is actually hit with the golf club head. Therefore thesimulation method is capable of accurately estimating the performanceand deformation behavior of the golf ball model in a state close to anactual hitting state. Thereby the simulation method is capable of easilyapprehending the property of the material which determines theperformance of the golf ball, thus contributing to the improvement ofthe product and reducing the number of trial manufactures of the golfball in a designing stage and cost and time required for the trialmanufacture.

1. A simulation method of estimating performance of a product made of aviscoelastic material, comprising the steps of: momently measuring avalue of each of a strain, a strain speed, and a stress generated insaid viscoelastic material in a measuring condition equivalent to acondition in which said product composed of said viscoelastic materialis actually used; deriving time history data of a viscous drag of saidviscoelastic material from time history data of each of said strain,said strain speed, and said stress and a viscoelastic model in which aviscosity of said viscoelastic material is considered, thereby derivinga relationship among said strain, said strain speed, and said viscousdrag; in estimating the performance of said product model whoseperformance is analyzed and made of said viscoelastic material saidproduct model is divided into a large number of elements saidrelationship is inputted to said product model, and an analysis isexecuted by a finite element method in consideration of a change of saidviscous drag in dependence on a variation of said strain and said strainspeed; and resolving said strain and said strain speed of an entirecoordinate system generated in each element into deviation componentsand volume components, and converting said strain and said strain speedof said deviation components from said entire coordinate system into amain strain coordinate system and a main strain speed coordinate systemrespectively; and determining a viscous drag for a coordinate axis ofeach of said main strain coordinate system and said main strain speedcoordinate system by using a converted deviation main strain and aconverted deviation main strain speed.
 2. The simulation method ofestimating performance of a product made of a viscoelastic materialaccording to claim 1, wherein said viscoelastic material showsnonlinearity in a property thereof.
 3. The simulation method ofestimating performance of a product made of a viscoelastic materialaccording to claim 1, wherein a split Hopkinson rod testing machinemeasures said strain, said strain speed, and said stress.
 4. Thesimulation method of estimating performance of a product made of aviscoelastic material according to claim 1, wherein a maximum value ofsaid strain generated in said viscoelastic material at said measuringtime is in a range of 0.05 to 0.50 and/or a maximum value of said strainspeed is in a range of 500/s to 10000/s.
 5. The simulation method ofestimating performance of a product made of a viscoelastic materialaccording to claim 1, wherein said viscoelastic material is used for agolf ball, and said product model is a golf ball.
 6. The simulationmethod of estimating performance of a product made of a viscoelasticmaterial according to claim 5, wherein a phenomenon of a collisionbetween said golf ball and a hitting object assumed to be a golf clubhead is simulated to estimate a behavior of said golf ball at the timeof said collision.
 7. A simulation method of estimating performance of aproduct made of a viscoelastic material, comprising the steps of:momently measuring a value of each of a strain, a strain speed, and astress generated in said product made of a viscoelastic materialequivalent to a condition in which said product composed of saidviscoelastic material is actually used; deriving a plurality ofdifferent rigidities from time history data of each of said strain andsaid stress and deriving a correspondence relationship among saidstrain, said strain speed, and said rigidities, thereby deriving arelationship among said strain, said strain speed, and said rigidities;in estimating the performance of said product model whose performance isanalyzed and made of a viscoelastic material said product model isdivided into a large number of elements, said relationship is inputtedto said product model, and an analysis is executed by a finite elementmethod in consideration of a change of said rigidities in dependence ona variation of said strain and said strain speed; and resolving saidstrain and said strain speed of an entire coordinate system generated ineach element into deviation components and volume components, andconverting said strain and said strain speed of said deviationcomponents from said entire coordinate system into a main straincoordinate system and a main strain speed coordinate systemrespectively; and determining a rigidity for a coordinate axis of eachof said main strain coordinate system and said main strain speedcoordinate system by using a converted deviation main strain and aconverted deviation main strain speed.
 8. The simulation method ofestimating performance of a product made of a viscoelastic materialaccording to claim 7, wherein time history data of a viscous drag ofsaid viscoelastic material is derived from time history data of each ofsaid strain, said strain speed, and said stress and a viscoelastic modelin which a viscosity of said viscoelastic material is considered; and achange of said rigidity and that of said viscous drag are considered. 9.The simulation method of estimating performance of a product made of aviscoelastic material according to claim 7, wherein said viscoelasticmaterial shows nonlinearity in a property thereof.
 10. The simulationmethod of estimating performance of a product made of a viscoelasticmaterial according to claim 7, wherein a split Hopkinson rod testingmachine measures said strain, said strain speed, and said stress. 11.The simulation method of estimating performance of a product made of aviscoelastic material according to claim 7, wherein a maximum value ofsaid strain generated in said viscoelastic material at said measuringtime is in a range of 0.05 to 0.50 and/or a maximum value of said strainspeed is in a range of 500/s to 10000/s.
 12. The simulation method ofestimating performance of a product made of a viscoelastic materialaccording to claim 7, wherein said viscoelastic material is used for agolf ball, and said product model is a golf ball.
 13. The simulationmethod of estimating performance of a product made of a viscoelasticmaterial according to claim 12, wherein a phenomenon of a collisionbetween said golf ball and a hitting object assumed to be a golf clubhead is simulated to estimate a behavior of said golf ball at a time ofsaid collision.
 14. A simulation method of estimating performance of aproduct made of a viscoelastic material, comprising the steps of:momently measuring a value of each of a strain, a strain speed, and astress generated in said product made of a viscoelastic material in ameasuring condition equivalent to a condition in which said productcomposed of said viscoelastic material is actually used; deriving timehistory data of a viscous drag of said viscoelastic material separatelyin a strain increase state and a strain decrease state or a strainrestoration state from time history data of each of said strain, saidstrain speed, and said stress and a viscoelastic model in which aviscosity of said viscoelastic material is considered, thereby derivinga relationship among said strain, said strain speed, and said viscousdrag; in estimating the performance of said product model whoseperformance is analyzed and made of said viscoelastic material, saidproduct model is divided into a large number of elements, saidrelationship is inputted to said product model, and an analysis isexecuted by a finite element method in consideration of a difference insaid viscous drag between said strain increase state and said straindecrease state or said strain restoration state; and resolving saidstrain and said strain speed of an entire coordinate system generated ineach element into deviation components and volume components, andconverting said strain and said strain speed of said deviationcomponents from said entire coordinate system into a main straincoordinate system and a main strain speed coordinate systemrespectively; and determining a viscous drag different between saidstrain increase state and said strain decrease or said strainrestoration state for a coordinate axis of each of said main straincoordinate system and said main strain speed coordinate system whenstrains having an equal value are generated in said viscoelasticmaterial, by using a converted deviation main strain and a converteddeviation main strain speed.
 15. The simulation method of estimatingperformance of a product made of a viscoelastic material according toclaim 14, wherein a norm which is the magnitude of a main strain iscomputed for each of said elements, and said computed norm is comparedwith a norm of a previous step in a simulation to determine that saidmain strain is in said increase state when said norm has increased, andto determine that said main strain is in said decrease state when saidnorm has decreased.
 16. The simulation method of estimating performanceof a product made of a viscoelastic material according to claim 14,wherein said viscoelastic material shows nonlinearity in a propertythereof.
 17. The simulation method according to claim 14, wherein asplit Hopkinson rod testing machine measures said strain, said strainspeed, and said stress.
 18. The simulation method according to claim 14,wherein a maximum value of said strain generated in said viscoelasticmaterial at said measuring time is in a range of 0.05 to 0.50 and/or amaximum value of said strain speed is in a range of 500/s to 10000/s.19. The simulation method according to claim 14, wherein saidviscoelastic material is used for a golf ball, and said product model isa golf ball.
 20. The simulation method according to claim 19, wherein aphenomenon of a collision between said golf ball and a hitting objectassumed to be a golf club head is simulated to estimate a behavior ofsaid golf ball at a time of said collision.